Research Article On a Family of Multivariate Modified ...e Scientic World Journal bilateral generating functions for the family of multivariable polynomials generated by ( ) and given

Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 187452 12 pageshttpdxdoiorg1011552013187452

Research ArticleOn a Family of Multivariate Modified Humbert Polynomials

Rabia AktaG1 and Esra ErkuG-Duman2

1 Department of Mathematics Faculty of Science Ankara University Tandogan 06100 Ankara Turkey2Department of Mathematics Faculty of Science Gazi University Teknikokullar 06500 Ankara Turkey

Correspondence should be addressed to Rabia Aktas raktasscienceankaraedutr

Received 29 April 2013 Accepted 6 June 2013

Academic Editors F J Garcia-Pacheco and V Privman

Copyright copy 2013 R Aktas and E Erkus-Duman This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper attempts to present a multivariable extension of generalized Humbert polynomials The results obtained here includevarious families ofmultilinear andmultilateral generating functions miscellaneous properties and also some special cases for thesemultivariable polynomials

1 Introduction

The generalized Humbert polynomials are generated by [1]

(119862 minus 119898119909119905 + 119910119905119898)119901

=infin

sum119899=0

119875119899(119898 119909 119910 119901 119862) 119905119899 (1)

where 119898 is a positive integer and other parameters areunrestricted in general (see also [2 pages 77 86] and [34]) This definition includes many well-known special poly-nomials such as Humbert Louville Gegenbauer LegendreTchebycheff Pincherle and Kinney polynomials

In this paper we consider the following multivariableextension of the generalized Humbert polynomials which arecompletely different from the polynomials introduced in [5]This class of polynomials is generated by

(1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

=infin

sum1198991119899119903=0

Θ(120572)

1198991 119899119903

(x ym) 11990511989911sdot sdot sdot 119905119899119903

119903

(1003816100381610038161003816100381610038161003816100381610038161003816

119903

sum119894=1

(119898119894119909119894t119894minus 119910

119894119905119898119894119894)1003816100381610038161003816100381610038161003816100381610038161003816lt 1)

(2)

where x = (1199091 119909

119903) y = (119910

1 119910

119903)m = (119898

1 119898

119903)

and 119898119894(119894 = 1 2 119903) is a positive integer It follows from

(2) that

Θ(120572)

1198991 119899119903

(x ym)

=[11989911198981]

sum1198961=0

sdot sdot sdot[119899119903119898119903]

sum119896119903=0

(120572)1198991+sdotsdotsdot+119899119903minus(1198981minus1)1198961minussdotsdotsdotminus(119898119903minus1)119896119903

1198961 sdot sdot sdot 119896

119903 (119899

1minus 119898

11198961) sdot sdot sdot (119899

119903minus 119898

119903119896119903)

times (minus1)1198961+sdotsdotsdot+119896119903(11989811199091)1198991minus11989811198961 sdot sdot sdot (119898

119903119909119903)119899119903minus1198981199031198961199031199101198961

1sdot sdot sdot 119910119896119903

119903

(3)

where (120582)119896

= 120582(120582 + 1) sdot sdot sdot (120582 + 119896 minus 1) (120582)0= 1 is the

Pochhammer symbolThe aim of this paper is to derive various families of

multilinear and multilateral generating functions and to giveseveral recurrence relations and expansions in the seriesof orthogonal polynomials for the family of multivariablepolynomials given explicitly by (3) We present some specialcases of our results and also obtain some other properties forthese special cases

2 Bilinear and Bilateral Generating Functions

In this section with the help of the similar method asconsidered in [5ndash9] we derive several families of bilinear and

2 The Scientific World Journal

bilateral generating functions for the family of multivariablepolynomials generated by (2) and given explicitly by (3)

We begin by stating the following theorem

Theorem 1 Corresponding to an identically nonvanishingfunction Ω

120583(z) of 119904 complex variables 119911

1 119911

119904(119904 isin N) and

of complex order 120583 let

Λ120583] (z 119908) =

infin

sum119896=0

119886119896Ω120583+]119896 (z) 119908

119896 (4)

where (119886119896

= 0 120583 ] isin C) z = (1199111 119911

119904)Then for119901 isin N x =

(1199091 119909

119903) y = (y

1 yr) m = (119898

1 119898

119903) 119898

119894isin N (119894 =

1 2 119903) one has

infin

sum1198991 119899119903=0

[1198991119901]

sum119896=0

119886119896Θ(120572)

1198991minus1199011198961198992119899119903

(x ym)

times 11990511989922sdot sdot sdot 119905119899119903

119903Ω120583+]119896 (z) 120578

1198961199051198991minus1199011198961

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

Λ120583] (z 120578)

(5)

provided that each member of (5) exists

Proof For convenience let 119878 denote the first member of theassertion (5) of Theorem 1 Replacing 119899

1by 119899

1+ 119901119896 we may

write that

119878 =infin

sum1198991 119899119903=0

infin

sum119896=0

119886119896Θ(120572)

11989911198992 119899119903

(x ym)Ω120583+]119896 (z) 120578

11989611990511989911sdot sdot sdot 119905119899119903

119903

=infin

sum1198991 119899119903=0

Θ(120572)

1198991 1198992119899119903


119903

infin

sum119896=0

119886119896Ω120583+]119896 (z) 120578

119896

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

Λ120583] (z 120578)

(6)

which completes the proof

By using a similar idea we also get the next result imme-diately

Theorem 2 For a nonvanishing function 1205931205821 120582119903

(z) of 119904complex variables 119911

1 119911

119904(119904 isin N) and for 119901

119894isin N 120582

119894 120578

119894isin C

z = (1199111 119911

119904) w = (119908

1 119908

119903) 120582 = (120582

1 120582

119903) 120578 =

(1205781 120578

119903) let

Ωnp120582120578120572120573m (x y zw)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

Θ(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(x ym)

times 1205931205821+12057811198961 120582119903+120578119903119896119903

(z) 1199081198961

1sdot sdot sdot 119908119896119903

119903(7)

where 1198861198961 119896119903

= 0 119899119894 (119894 = 1 2 119903) isin N

0 N

0= N cup 0

Then one has

1198991

sum1198971=0

sdot sdot sdot119899119903

sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

Θ(120572)

1198991minus1198971 119899119903minus119897119903

(x ym)

times Θ(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(x ym)

times 1205931205821+12057811198961 120582r+120578119903119896119903

(z)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Ωnp120582120578120572120573m (x y zw)

(8)


3 Special Cases

As an application of the above theorems when the multi-variable function Ω

120583+]119896(z) z = (1199111 119911

119904) 119896 isin N

0 119904 isin N

is expressed in terms of simpler functions of one and morevariables then we can give further applications of the abovetheorems We first set

119904 = 119903 Ω120583+]119896 (z) = g(1205731 120573119903)

120583+]119896 (z) (9)

in Theorem 1 where the Chan-Chyan-Srivastava polynomi-als g(1205731 120573119903)

120583+]119896 (z) [10] are generated by

119903

prod119894=1

(1 minus 119909119894119905)minus120572119894 =

infin

sum119899=0

g(1205721120572119903)119899

(1199091 119909

119903) 119905119899

(120572119894isin C (119894 = 1 2 119903) |119905| lt min 10038161003816100381610038161199091

1003816100381610038161003816minus1

10038161003816100381610038161199091199031003816100381610038161003816minus1

) (10)

We are thus led to the following result which provides aclass of bilateral generating functions for the Chan-Chyan-Srivastava polynomials and the family of multivariable poly-nomials given explicitly by (3)

Corollary 3 If Λ120583](z 119908) = suminfin

119896=0119886119896g (1205731 120573119903)

120583+]119896 (z)119908119896 119886119896

= 0120583 ] isin C z = (119911

1 119911

119903) then one has

infin

sum1198991 119899119903=0

[1198991119901]

sum119896=0

119886119896Θ(120572)

1198991minus1199011198961198992 119899119903

(x ym)


119903g (1205731 120573119903)

120583+]119896 (z) 1205781198961199051198991minus1199011198961

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

Λ120583] (z 120578)

(11)


The Scientific World Journal 3

Remark 4 Using the generating relation (10) for the Chan-Chyan-Srivastava polynomials and getting 119886

119896= 1 120583 = 0 ] =

1 we find that

infin

sum1198991 119899119903=0

[1198991119901]

sum119896=0

Θ(120572)

1198991minus1199011198961198992 119899119903


119903g(1205731 120573119903)119896

(z)

times 1205781198961199051198991minus1199011198961

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572119903

prod119894=1

(1 minus 119911119894120578)

minus120573119894

10038161003816100381610038161205781003816100381610038161003816 lt min 10038161003816100381610038161199111

1003816100381610038161003816minus1

10038161003816100381610038161199111199031003816100381610038161003816minus1

1003816100381610038161003816100381610038161003816100381610038161003816

119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894)1003816100381610038161003816100381610038161003816100381610038161003816lt 1

(12)

On the other hand choosing 119904 = 2119903 and

1205931205821+12057811198961 120582119903+120578119903119896119903

(z) = Θ(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(u km) (13)

120582119894 120578

119894(119894 = 1 2 119903) isin N

0 u = (119906

1 119906

119903) k = (V

1

V119903) inTheorem 2 we obtain the following class of bilinear

generating functions for the family of multivariable polyno-mials given explicitly by (3)

Corollary 5 If

Λnp120582120578120572120573120574m (x y u kw)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

Θ(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(x ym)

times Θ(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(u km) 1199081198961

1sdot sdot sdot 119908119896119903

119903

(14)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 then

we get

1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

Θ(120572)

1198991minus1198971 119899119903minus119897119903

(x ym)

times Θ(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(x ym)

timesΘ(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(u km)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Λnp120582120578120572120573120574m (x y u kw)

(15)


Furthermore for every suitable choice of the coefficients119886119896

(119896 isin N0) if the multivariable functions Ω

120583+]119896(y) and1205931205821+12057811198961 120582119903+120578119903119896119903

(y) y = (1199101 119910

119904) (119904 isin N) are expressed

as an appropriate product of several simpler functions theassertions of Theorems 1 and 2 can be applied in orderto derive various families of multilinear and multilateral


4 Some Miscellaneous Properties

In this section we now discuss some further properties ofthe family of multivariable polynomials given by (3)We startwith the following theorems

Theorem 6 Let Ψ1198991119899119903

(x ym) be a family of functionsgenerated by

119866 (119898111990911199051minus 119910

111990511989811 119898

119903119909119903119905119903minus 119910

119903119905119898119903119903)

=infin

sum1198991 119899119903=0

Ψ1198991119899119903


119903

(16)

The following relations hold

119899119894Ψ1198991119899119903

(x ym) = 119909119894

120597

120597119909119894

Ψ1198991119899119903

(x ym)

minus 119910119894

120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

(17)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

119899119894Ψ1198991119899119903

(x ym) = 119909119894

120597

120597119909119894

Ψ1198991119899119903

(x ym) (18)

for 119899119894= 0 1 119898

119894minus 2 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 Also one

finds that

119898119894119909119894

120597

120597119910119894

Ψ1198991119899119903

(x ym) minus 119898119894119910119894

120597

120597119910119894

times Ψ1198991119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(x ym)

= minus (119899119894minus 119898

119894+ 1)Ψ

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903(x ym)

(19)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Ψ1198991119899119903

(x ym) = 0 (20)

for 119899119894= 0 1 119898

119894minus 2 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


Proof Fix 119894 = 1 2 119903 Then by differentiating (16) withrespect to 119909

119894and 119905

119894 after making necessary calculations we

obtain thatinfin

sum1198991119899119903=0

119899119894Ψ1198991119899119903


119903

= (119909119894minus 119910

119894119905119898119894minus1119894

)infin

sum1198991119899119903=0

120597

120597119909119894

Ψ1198991119899119903


119903

= 119909119894

infin

sum1198991 119899119903=0

120597

120597119909119894

Ψ1198991119899119903


119903

minus 119910119894

infin

sum1198991 119899119894minus1119899119894+1119899119903=0

(119899119894ge119898119894minus1)

120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)


119903

(21)

Comparing the coefficients of 11990511989911sdot sdot sdot 119905119899119903

119903 we obtain (17) and

(18) for the fixed 119894 Similarly if we differentiate (16) withrespect to 119910

119894and 119905

119894 we can find the relation (19)

Theorem 7 If Ψ1198991119899119903

(x ym) is a family of functions gener-ated by (16) then it satisfies the relations

minus 119898119894

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1 119899119903

(x ym)

=120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894 119899119894+1119899119903

(x ym)

(22)

for 119899119894ge 119898

119894 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1119899119903

(x ym) = 0 (23)

for 119899119894= 1 2 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894

Proof By comparing the derivatives of (16) with respect to 119909119894

and 119910119894 we have

minus 119898119894119905119894

infin

sum1198991 119899119903=0

120597

120597119910119894

Ψ1198991119899119903


119903

= 119905119898119894119894

infin

sum1198991 119899119903=0

120597

120597119909119894

Ψ1198991 119899119903


119903

(24)

which implies that

minus 119898119894

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1 119899119903

(x ym)

=120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894 119899119894+1119899119903

(x ym)

(25)

for 119899119894ge 119898

119894 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1119899119903

(x ym) = 0 (26)

for 119899119894= 1 2 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 Thus

the proof is completed

As a result of these theorems if we choose

119866 (119898111990911199051minus 119910

111990511989811 119898

119903119909119903119905119903minus 119910

119903119905119898119903119903)

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

(27)

then we can give some recurrence relations for the family ofmultivariable polynomials given explicitly by (3) In the viewof Theorem 6 we get

Corollary 8 For the family of multivariable polynomialsgenerated by (2) the following relations hold

119899119894Θ(120572)

1198991 119899119903

(x ym) = 119909119894

120597

120597119909119894

Θ(120572)

1198991119899119903

(x ym)

minus 119910119894

120597

120597119909119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(x ym)

(28)

119898119894119909119894

120597

120597119910119894

Θ(120572)

1198991 119899119903

(x ym) minus 119898119894119910119894

120597

120597119910119894

times Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(x ym)

= minus (119899119894minus 119898

119894+ 1)Θ(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

(29)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 Also for 119899

119894=

0 1 119898119894minus 2 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 we have

119899119894Θ(120572)

1198991 119899119903

(x ym) = 119909119894

120597

120597119909119894

Θ(120572)

1198991119899119903

(x ym) (30)

120597

120597119910119894

Θ(120572)

1198991119899119903

(x ym) = 0 (31)

Corollary 9 By summing the relations given by (28) and (29)respectively for 119894 = 1 2 119903 we get

119903

sum119894=1

119899119894Θ(120572)

1198991119899119903

(x ym)

=119903

sum119894=1

119909119894

120597

120597119909119894

Θ(120572)

1198991119899119903

(x ym)

minus119903

sum119894=1

119910119894

120597

120597119909119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(x ym)


119903

sum119894=1

119898119894119909119894

120597

120597119910119894

Θ(120572)

1198991 119899119903

(x ym)

minus119903

sum119894=1

119898119894119910119894

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

= minus119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

(32)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894

Similarly as a consequence ofTheorem 7 we can give thenext result at once

Corollary 10 Other recurrence relations for the family ofmultivariable polynomials Θ(120572)

1198991119899119903

(x ym) are

minus 119898119894

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus1119899119894+1 119899119903

(x ym)

=120597

120597119909119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894 119899119894+1119899119903

(x ym)

(33)

for 119899119894ge 119898

119894 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(x ym) = 0 (34)

for 119899119894= 1 2 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894

Theorem 11 The generating relation (2) yields the followingaddition formula for the family of multivariable polynomialsgiven by (3)

Θ(120572+120573)

1198991119899119903

(x ym) =1198991

sum1198961=0


sum119896119903=0

Θ(120572)

1198991minus1198961 119899119903minus119896119903

(x ym)

times Θ(120573)

1198961 119896119903

(x ym)

(35)

Proof From (2) we have

infin

sum1198991 119899119903=0

Θ(120572+120573)

1198991119899119903


119903

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus(120572+120573)

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

times (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120573

=infin

sum1198991 119899119903=0

Θ(120572)

1198991 119899119903


119903

timesinfin

sum1198961 119896119903=0

Θ(120573)

1198961 119896119903


119903

(36)

Replacing 119899119894by 119899

119894minus 119896

119894 119894 = 1 2 119903 the right-hand side of

the last equality is

infin

sum1198991119899119903=0

1198991

sum1198961=0


sum119896119903=0

Θ(120572)

1198991minus1198961 119899119903minus119896119903

(x ym)

times Θ(120573)

1198961 119896119903


119903

(37)


119903completes the proof

We now give expansions of the family of multivariablepolynomialsΘ(120572)

1198991119899119903

(x ym) given explicitly by (3) in series ofLegendre Gegenbauer Hermite and Laguerre polynomials

Theorem 12 Expansions of Θ(120572)

1198991119899119903

(x ym) in series of Leg-endre Gegenbauer Hermite and Laguerre polynomials are asfollows

Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119896119894119904119894(32)

119899119894minus119898119894119896119894minus119904119894

times(minus1)119896119894119910119896119894119894119875119899119894minus119898119894119896119894minus2119904119894

(119898119894119909119894

2)

times (120572)1198991+sdotsdotsdot+119899119903minus(1198981minus1)1198961minussdotsdotsdotminus(119898119903minus1)119896119903

Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(]119894+ 119899

119894minus 119898

119894119896119894minus 2119904

119894)

119896119894119904119894(]

119894)119899119894minus119898119894119896119894minus119904119894+1

times(minus119910119894)119896119894119862]119894

119899119894minus119898119894119896119894minus2119904119894

(119898119894119909119894

2)


Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898i]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(minus119910119894)119896119894119867

119899119894minus119898119894119896119894minus2119904119894(119898

1198941199091198942)

119896119894119904119894 (119899

119894minus 119898

119894119896119894minus 2119904

119894)



Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

119899119894minus119898119894119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus119898119894119896119894

2119899119894minus119898119894119896119894(minus1)119904119894

119896119894 (119899

119894minus 119898

119894119896119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus119910119894)119896119894119871(120573119894)

119904119894

(119898119894119909119894

2)


(38)

Proof By (2) we get

infin

sum1198991119899119903=0

Θ(120572)

1198991 119899119903


119903

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

=infin

sum1198991 119899119903=0

infin

sum1198961 119896119903=0

(120572)1198991+sdotsdotsdot+119899119903+1198961+sdotsdotsdot+119896119903


1199031198991 sdot sdot sdot 119899

119903(minus1)1198961+sdotsdotsdot+119896119903

times (11989811199091)1198991 sdot sdot sdot (119898

119903119909119903)1198991199031199101198961

1sdot sdot sdot 119910119896119903

119903

times 1199051198991+119898111989611

sdot sdot sdot 119905119899119903+119898119903119896119903119903

(39)

If we use the result in [11 page 181]

(119898119909)119899

119899=

[1198992]

sum119904=0

(2119899 minus 4119904 + 1) 119875119899minus2119904

(1198981199092)

119904(32)119899minus119904

(40)

we can write that

infin

sum1198991119899119903=0

Θ(120572)

1198991 119899119903


119903

=119903

prod119894=1

infin

sum119899119894=0

infin

sum119896119894=0

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119898119894119909119894

2)

times(minus119910119894)119896119894119905119899119894+119898119894119896119894

119894


(41)

Getting 119899119894minus 119898

119894119896119894instead of 119899

119894in the last equality we have

infin

sum1198991 119899119903=0

Θ(120572)

1198991119899119903


119903

=infin

sum1198991 119899119903=0

119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119898119894119896119894minus119904119894

times119875119899119894minus119898119894119896119894minus2119904119894

(119898119894119909119894

2) (minus119910

119894)119896119894119905119899119894

119894


=infin

sum1198991 119899119903=0

119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2) (minus119910

119894)119896119894


11990511989911sdot sdot sdot 119905119899119903

119903

(42)


119903gives the desired

relationIn a similar manner in (39) using the following results

respectively [11 page 283 (36) page 194 (4) page 207 (2)]

(119898119909)119899

119899=

[1198992]

sum119896=0

(] + 119899 minus 2119896) 119862]119899minus2119896

(1198981199092)

119896(])119899+1minus119896

(119898119909)119899

119899=

[1198992]

sum119896=0

119867119899minus2119896

(1198981199092)

119896 (119899 minus 2119896)

(119898119909)119899

119899= 2119899

119899

sum119896=0

(minus1)119896(120572 + 1)119899119871(120572)119896

(1198981199092)

(119899 minus 119896)(120572 + 1)119896

(43)

one can easily obtain the other expansions ofΘ(120572)

1198991 119899119903

(x ym)in series of Gegenbauer Hermite and Laguerre polynomials

5 The Special Cases of Θ(120572)

119899(x y m) and

Some Properties

In this section we discuss some special cases of the family ofmultivariable polynomialsΘ(120572)

119899(x ym) and give their several

properties

51TheCase of119898119894= 2 119910

119894= 1 119894 = 1 2 119903 in (2) This case

gives a multivariable analogue of Gegenbauer polynomials

(1 minus 211990911199051+ 1199052

1minus sdot sdot sdot minus 2119909

119903119905119903+ 1199052

119903)minus120572

=infin

sum1198991 119899119903=0

119862(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(44)


which reduces to two variables Gegenbauer polynomialsgiven by [12] for 119903 = 2 Equation (44) yields the followingformula

119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0

sdot sdot sdot[1198991199032]

sum119896119903=0

(((120572)1198991+sdotsdotsdot+119899119903minus1198961minussdotsdotsdotminus119896119903

)

times (1198961 sdot sdot sdot 119896

119903 (119899

1minus 2119896

1)

sdot sdot sdot (119899119903minus 2119896

119903))

minus1

)


119903)119899119903minus2119896119903

(45)

It follows that 119862(120572)

1198991 119899119903

(1199091 119909

119903) is a polynomial of

degree 119899119894with respect to the fixed variable 119909

119894(119894 = 1 2 119903)

Thus 119862(120572)

1198991 119899119903

(1199091 119909

119903) is a polynomial of total degree (119899

1+

sdot sdot sdot+119899119903)with respect to the variables 119909

1 119909

119903 Equation (45)

also yields

119862(120572)

1198991 119899119903

(1199091 119909

119903)

=21198991+sdotsdotsdot+119899119903(120572)

1198991+sdotsdotsdot+119899119903


119903

11990911989911sdot sdot sdot 119909119899119903

119903+ Π (119909

1 119909

119903)

(46)

whereΠ(1199091 119909

119903) is a polynomial of degree (119899

1+ sdot sdot sdot + 119899

119903minus

2) with respect to the variables 1199091 119909

119903 In (44) by getting

1199091rarr minus119909

1and 119905

1rarr minus119905

1 we have

119862(120572)

1198991 119899119903

(minus1199091 119909

2 119909

119903) = (minus1)1198991119862(120572)

1198991 119899119903

(1199091 119909

119903)

(47)

Similarly for 119894 = 1 2 119903 we get

119862(120572)

1198991 119899119903

(1199091 119909

119894minus1 minus119909

119894 119909

119894+1 119909

119903)

= (minus1)119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

(48)

Taking 119909119894rarr minus119909

119894and 119905

119894rarr minus119905

119894 119894 = 1 2 119903 in (44)

we obtain

119862(120572)

1198991 119899119903

(minus1199091 minus119909

119903) = (minus1)1198991+sdotsdotsdot+119899119903119862(120572)

1198991 119899119903

(1199091 119909

119903)

(49)

Theorem 13 For the polynomials119862(120572)

1198991 119899119903

(1199091 119909

119903) one has

119862(120572)

21198991 2119899119903

(0 0 0) =(minus1)1198991+sdotsdotsdot+119899119903(120572)

1198991+sdotsdotsdot+119899119903


119903

(50)

and if at least one of 119899119894 119894 = 1 2 119903 is odd then

119862(120572)

1198991 119899119903

(0 0 0) = 0 (51)

Proof If we set all 119909119894= 0 119894 = 1 2 119903 in (44) we have

(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991119899119903=0

119862(120572)

1198991 119899119903

(0 0 0) 11990511989911sdot sdot sdot 119905119899119903

119903

(52)

On the other hand we get

(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991 119899119903=0

(minus1)1198991+sdotsdotsdot+119899119903


119903(120572)

1198991+sdotsdotsdot+119899119903119905211989911

sdot sdot sdot 1199052119899119903119903

(53)

By comparing the coefficients of 11990511989911sdot sdot sdot 119905119899119903

119903 we obtain the

desired

From the theorems and corollaries given in Section 4 wecan give some other properties of 119862(120572)

1198991 119899119903

(1199091 119909

119903)

Remark 14 ByCorollaries 8 and 9 for the family ofmultivari-able polynomials generated by (44) the following relations

119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

= 119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus119903

sum119894=1

120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

(54)

hold for 119899119894ge 1 119894 = 1 2 119903

Remark 15 FromTheorem 11 the multivariable polynomials119862(120572)

1198991 119899119903

(1199091 119909

119903) satisfy the following addition formula

119862(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119862(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119862(120573)

1198961 119896119903

(1199091 119909

119903)

(55)


Remark 16 As a result of Theorem 12 expansions of 119862(120572)

1198991 119899119903

(1199091 119909

119903) in series of Legendre Gegenbauer Hermite and

Laguerre polynomials are as follows

119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(2119899119894minus 4119896

119894minus 4119904

119894+ 1)

119896119894119904119894(32)

119899119894minus2119896119894minus119904119894


(119909119894)

times (120572)1198991+sdotsdotsdot+119899119903minus1198961minussdotsdotsdotminus119896119903

119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(]119894+ 119899

119894minus 2119896

119894minus 2119904

119894)

119896119894119904119894(]

119894)119899119894minus2119896119894minus119904119894+1

times(minus1)119896119894119862]119894119899119894minus2119896119894minus2119904119894

(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(minus1)119896119894119867119899119894minus2119896119894minus2119904119894

(119909119894)

119896119894119904119894 (119899

119894minus 2119896

119894minus 2119904

119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

119899119894minus2119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus2119896119894

2119899119894minus2119896119894(minus1)119904119894

119896119894 (119899

119894minus 2119896

119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus1)119896119894119871(120573119894)119904119894

(119909119894)


(56)

We now give a hypergeometric representation for themultivariable polynomials 119862(120572)

1198991 119899119903

(1199091 119909

119903) given by (44)

Theorem 17 The multivariable polynomials 119862(120572)

1198991 119899119903

(1199091

119909119903) have the following hypergeometric representation

119862(120572)

1198991 119899119903

(1199091 119909

119903)

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

times 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2

1 minus 120572 minus 1198991minus sdot sdot sdot minus 119899

1199031

11990921

1

1199092119903

]

(max 1

11990921

1

1199092119903

lt 1)

(57)

where 119865(119903)119861

is a Lauricella function of 119903-variable defined by [13](see also [2])

119865(119903)119861

[1198861 119886

119903 1198871 119887

119903 119888 119909

1 119909

119903]

=infin

sum1198991119899119903=0

(1198861)1198991

sdot sdot sdot (119886119903)119899119903

(1198871)1198991

sdot sdot sdot (119887119903)119899119903

(119888)1198991+sdotsdotsdot+119899119903

11990911989911

1198991sdot sdot sdot

119909119899119903119903

119899119903

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091199031003816100381610038161003816 lt 1)

(58)

Proof We have the following results from [11]

(120572)119899minus119896

=(minus1)119896(120572)

119899

(1 minus 120572 minus 119899)119896

(minus119899)2119896

= 22119896(minus119899

2)119896

(minus119899 + 1

2)119896

(119899 minus 2119896) =119899

(minus119899)2119896

(59)

In view of these relations the equality (45) can be written asfollows

119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0

(((120572)1198991+sdotsdotsdot+119899119903

22(1198961+sdotsdotsdot+119896119903)(minus1198991

2)1198961

times (minus119899

1+ 1

2)1198961

sdot sdot sdot (minus119899119903

2)119896119903

times(minus119899

119903+ 1

2)119896119903

)


1199031198991 sdot sdot sdot 119899

119903

times (1 minus 120572 minus 1198991minus sdot sdot sdot minus 119899

119903)1198961+sdotsdotsdot+119896119903

)minus1

)

times (21199091)1198991minus21198961 sdot sdot sdot (2119909

119903)119899119903minus2119896119903

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903


times[11989912]

sum1198961=0


sum119896119903=0

(((minus1198991

2)1198961

(minus119899

1+ 1

2)1198961


2)119896119903

(minus119899

119903+ 1

2)119896119903

)

times ((1 minus 120572 minus 1198991minus sdot sdot sdot minus 119899

119903)1198961+sdotsdotsdot+119896119903

times 1198961 sdot sdot sdot 119896

119903)

minus1

)

times(1

11990921

)1198961

sdot sdot sdot (1

1199092119903

)119896119903

= 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

times(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

(60)

wheremax 111990921 11199092

119903 lt 1The proof is completed

Remark 18 For 119903 = 2 Theorem 17 reduces to the knownresult for two variable Gegenbauer polynomials given by [12]

52 The Case of 119909119894= 0 119910

119894= minus119909

119894 119894 = 1 2 119903 in (2) This

case yields

(1 minus 119909111990511989811

minus 119909211990511989822

minus sdot sdot sdot minus 119909119903119905119898119903119903)minus120572

=infin

sum1198991 119899119903=0

119880(120572)

1198991119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(61)

which is a different unification from that in [8]

Remark 19 FromCorollaries 8 and 9 the multivariable poly-nomials 119880(120572)

1198991119899119903

(1199091 119909

119903) satisfy

119898119894119909119894

120597

120597119909119894

119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

= (119899119894minus 119898

119894+ 1)119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119898119894119909119894

120597

120597119909119894

times 119880(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times 119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

(62)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894

Remark 20 As result of Theorem 11 the multivariable poly-nomials 119880(120572)

1198991 119899119903

(1199091 119909

119903) have the following addition for-

mula

119880(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119880(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119880(120573)

1198961 119896119903

(1199091 119909

119903)

(63)

521 The Case of 119898119894= 1 119894 = 1 2 119903 in Section 52 In

this case we get a 119903-variable analogue of Lagrange polyno-mials (see [14]) which is different from that in [10]

(1 minus 11990911199051minus sdot sdot sdot minus 119909

119903119905119903)minus120572

=infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(64)

They are given explicitly by

119866(120572)

1198991 119899119903

(1199091 119909

119903) =

(120572)1198991+sdotsdotsdot+119899119903


11990311990911989911sdot sdot sdot 119909119899119903

119903 (65)

Remark 21 From Corollaries 8 and 9 the multivariablepolynomials 119866(120572)

1198991 119899119903

(1199091 119909

119903) satisfy

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903) = 119909

119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903) (66)

for 119894 = 1 2 119903 and119903

sum119894=1

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903)

(67)

Remark 22 By Theorem 11 the multivariable polynomials119866(120572)

1198991 119899119903

(1199091 119909

119903) have the following addition formula

119866(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119866(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119866(120573)

1198961 119896119903

(1199091 119909

119903)

(68)


Remark 23 From Theorem 12 expansions of 119866(120572)

1198991 119899119903

(1199091


Laguerre polynomials are given by

119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119909119894

2)

times (120572)1198991+sdotsdotsdot+119899119903

119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(]119894+ 119899

119894minus 2119904

119894)

119904119894(]

119894)119899119894minus119904119894+1

119862]119894119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

119867119899119894minus2119904119894

(1199091198942)

119904119894 (119899

119894minus 2119904

119894) (120572)

1198991+sdotsdotsdot+119899119903

119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

119899119894

sum119904119894=0

(120573119894+ 1)

119899119894

2119899119894(minus1)119904119894

(119899119894minus 119904

119894)(120573

119894+ 1)

119904119894

119871(120573119894)119904119894

(119909119894

2)


(69)

We can discuss some generating functions for the multi-variable polynomials 119866(120572)

1198991 119899119903

(1199091 119909

119903)

Theorem 24 For the polynomials 119866(120572)

1198991 119899119903

(1199091 119909

119903) the

following generating function holds true for 119896 isin N0

infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903

= 119898(1 minus 11990911199111minus sdot sdot sdot minus 119909

119903119911119903)minus120572

times g (120572minus119896)

119898(

119909119894119911119894

1 minus 11990911199111minus sdot sdot sdot minus 119909

119903119911119903

minus1)

(70)

for each 119894 = 1 2 119903 where

[119896]119898= 119896 (119896 minus 1) sdot sdot sdot (119896 minus 119898 + 1) 119898 isin N [119896]

0= 1 (71)

and g (120572120573)

119899(119909 119910) denotes the classical Lagrange polynomials

defined by [14]

(1 minus 119909119905)minus120572(1 minus 119910119905)minus120573

=infin

sum119899=0

g (120572120573)

119899(119909 119910) 119905119899

(120572 120573 isin C |119905| lt min |119909|minus1 10038161003816100381610038161199101003816100381610038161003816minus1

)

(72)

Proof Fix 119894 = 1 2 119903 LetC120585119894andClowast

120585119894

denote the circles ofradius 120576

119894 centered 120585

119894= 119911

119894and 120585

119894= 0 respectively where

0 lt 120576119894lt 1003816100381610038161003816119909119894

1003816100381610038161003816minus1

10038161003816100381610038161003816100381610038161003816100381610038161 minus

119903

sum119896=1

119909119896119911119896

1003816100381610038161003816100381610038161003816100381610038161003816 (73)

Here these circles are described in the positive direction ByCauchyrsquos Integral Formula we have

119863119898

119911119894(1 minus 119909


119903119911119903)minus120572

119911119896

119894

=

119898

2120587119894

∮C120585119894

120585119896

119894(1 minus 119909



119903119911119903)minus120572

(120585119894minus 119911

119894)119898+1

119889120585119894

=

119898

2120587119894

times∮Clowast120585119894

(120585119894+ 119911

119894)119896


119894(120585119894+ 119911

119894) minus sdot sdot sdot minus 119909

119903119911119903)minus120572

120585119898+1

119894

119889120585119894

=

119898

2120587119894

119911119896

119894(1 minus 119909


119903119911119903)minus120572


(1 + 120585119894119911

119894)119896

120585119898+1

119894

(1 minus

119909119894120585119894


119903119911119903

)

minus120572

119889120585119894

= 119898119911119896

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894


119903119911119903

minus

1

119911119894

)

= 119898119911119896minus119898

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894119911119894


119903119911119903

minus1)

(74)

On the other hand we can write that

119863119898

119911119894


119903119911119903)minus120572

119911119896119894

= 119863119898

119911119894

119911119896119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903

=infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903)119889119898

119889119911119898119894

11991111989911sdot sdot sdot 119911119899119894+119896

119894sdot sdot sdot 119911119899119903

119903

= 119911119896minus119898119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903

(75)

From (74) and (75) we see that

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572

times g(120572minus119896)119898

(119909119894119911119894


119903119911119903

minus1)

(76)

Corollary 25 From (71) one observes that

[119899119894+ 119898]

119898= (119899

119894+ 119898) (119899

119894+ 119898 minus 1) sdot sdot sdot (119899

119894+ 1)

= (119899119894+ 1)

119898

(77)


for each 119894 = 1 2 119903 By setting 119896 = 119898 in (70) and then using(77) one has

infin

sum1198991 119899119903=0

(119899119894+ 1)

119898119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(78)

522TheCase of119898119894= 119894 119894 = 1 2 119903 in Section 52 This case

reduces to a multivariable analogue of Lagrange-Hermitepolynomials

(1 minus 11990911199051minus 119909


119903119905119903119903)minus120572

=infin

sum1198991119899119903=0

119867(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(79)

which is different from that in [6]

Remark 26 By Corollaries 8 and 9 the multivariable polyno-mials119867(120572)

1198991 119899119903

(1199091 119909

119903) satisfy

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

= (119899119894minus 119894 + 1)119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

119903

sum119894=1

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119894 + 1)

times 119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

(80)

for 119899119894ge 119894 minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


119867(120572+120573)

1198991119899119903

(1199091 119909

119903)

=1198991

sum1198961=0


sum119896119903=0

119867(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119867(120573)

1198961 119896119903

(1199091 119909

119903)

(81)

Furthermore setting 119898119894

= 119894 119909119894

= 0 119910119894

= minus119909119894

119894 = 1 2 119903 in Corollary 5 we obtain the followingclass of bilinear generating functions for the polynomials119867(120572)

1198991 119899119903

(1199091 119909

119903)

Remark 28 If

Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)1199081198961

1sdot sdot sdot 119908119896119903

119903

(82)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 and

120582 = (1205821 120582

119903) 120578 = (120578

1 120578

119903)w = (119908

1 119908

119903) then we

have1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572)

1198991minus1198971 119899119903minus119897119903

(1199091 119909

119903)

times 119867(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

(83)

References

[1] H W Gould ldquoInverse series relation and other expansionsinvolving Humbert polynomialsrdquo Duke Mathematical Journalvol 32 pp 697ndash711 1965

[2] H M Srivastava and H L A Manocha Treatise on GeneratingFunctions Halsted Press Ellis Horwood Limited John Wileyand Sons Chichester UK 1984

[3] J P Singhal ldquoA note on generalized Humbert polynomialsrdquoGlasnik Matematicki III vol 5 no 25 pp 241ndash245 1970

[4] D Zeitlin ldquoOn a class of polynomials obtained fromgeneralizedHumbert polynomialsrdquo Proceedings of the AmericanMathemat-ical Society vol 18 pp 28ndash34 1967

[5] R Aktas R Sahin and A Altın ldquoOn a multivariable extensionof Humbert polynomialsrdquo Applied Mathematics and Computa-tion vol 218 pp 662ndash666 2011

[6] A Altın and E Erkus ldquoOn a multivariable extension ofthe Lagrange-Hermite polynomialsrdquo Integral Transforms andSpecial Functions vol 17 no 4 pp 239ndash244 2006

[7] A Altın E Erkus and M A Ozarslan ldquoFamilies of linear gen-erating functions for polynomials in two variablesrdquo IntegralTransforms and Special Functions vol 17 no 5 pp 315ndash3202006

[8] E Erkus and H M Srivastava ldquoA unfied presentation of somefamilies of multivariable polynomialsrdquo Integral Transforms andSpecial Functions vol 17 pp 267ndash273 2006

[9] E Ozergin M A Ozarslan and H M Srivastava ldquoSome fami-lies of generating functions for a class of bivariate polynomialsrdquoMathematical and Computer Modelling vol 50 pp 1113ndash11202009

[10] W-C C Chan C-J Chyan and H M Srivastava ldquoTheLagrange polynomials in several variablesrdquo Integral Transformsand Special Functions vol 12 no 2 pp 139ndash148 2001


[11] E D Rainville Special Functions The Macmillan CompanyNew York NY USA 1960

[12] M A Khan and G S Kahmmash ldquoA study of a two variablesGegenbauer polynomialsrdquo Applied Mathematical Sciences vol2 no 13 pp 639ndash657 2008

[13] G Lauricella ldquoSulle funzioni ipergeometriche a piu variabilirdquoRendiconti del CircoloMatematico di Palermo vol 7 pp 111ndash1581893

[14] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 3 McGraw-Hill BookCompany New York NY USA 1955

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


bilateral generating functions for the family of multivariablepolynomials generated by (2) and given explicitly by (3)

We begin by stating the following theorem

Theorem 1 Corresponding to an identically nonvanishingfunction Ω

120583(z) of 119904 complex variables 119911

1 119911

119904(119904 isin N) and

of complex order 120583 let

Λ120583] (z 119908) =

infin

sum119896=0

119886119896Ω120583+]119896 (z) 119908

119896 (4)

where (119886119896

= 0 120583 ] isin C) z = (1199111 119911

119904)Then for119901 isin N x =

(1199091 119909

119903) y = (y

1 yr) m = (119898

1 119898

119903) 119898

119894isin N (119894 =

1 2 119903) one has

infin

sum1198991 119899119903=0

[1198991119901]

sum119896=0

119886119896Θ(120572)

1198991minus1199011198961198992119899119903

(x ym)


119903Ω120583+]119896 (z) 120578

1198961199051198991minus1199011198961

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

Λ120583] (z 120578)

(5)


Proof For convenience let 119878 denote the first member of theassertion (5) of Theorem 1 Replacing 119899

1by 119899

1+ 119901119896 we may

write that

119878 =infin

sum1198991 119899119903=0

infin

sum119896=0

119886119896Θ(120572)

11989911198992 119899119903

(x ym)Ω120583+]119896 (z) 120578

11989611990511989911sdot sdot sdot 119905119899119903

119903

=infin

sum1198991 119899119903=0

Θ(120572)

1198991 1198992119899119903


119903

infin

sum119896=0

119886119896Ω120583+]119896 (z) 120578

119896

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

Λ120583] (z 120578)

(6)

which completes the proof

By using a similar idea we also get the next result imme-diately

Theorem 2 For a nonvanishing function 1205931205821 120582119903

(z) of 119904complex variables 119911

1 119911

119904(119904 isin N) and for 119901

119894isin N 120582

119894 120578

119894isin C

z = (1199111 119911

119904) w = (119908

1 119908

119903) 120582 = (120582

1 120582

119903) 120578 =

(1205781 120578

119903) let

Ωnp120582120578120572120573m (x y zw)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

Θ(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(x ym)

times 1205931205821+12057811198961 120582119903+120578119903119896119903

(z) 1199081198961

1sdot sdot sdot 119908119896119903

119903(7)

where 1198861198961 119896119903

= 0 119899119894 (119894 = 1 2 119903) isin N

0 N

0= N cup 0

Then one has

1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

Θ(120572)

1198991minus1198971 119899119903minus119897119903

(x ym)

times Θ(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(x ym)

times 1205931205821+12057811198961 120582r+120578119903119896119903

(z)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Ωnp120582120578120572120573m (x y zw)

(8)


3 Special Cases

As an application of the above theorems when the multi-variable function Ω

120583+]119896(z) z = (1199111 119911

119904) 119896 isin N

0 119904 isin N

is expressed in terms of simpler functions of one and morevariables then we can give further applications of the abovetheorems We first set

119904 = 119903 Ω120583+]119896 (z) = g(1205731 120573119903)

120583+]119896 (z) (9)

in Theorem 1 where the Chan-Chyan-Srivastava polynomi-als g(1205731 120573119903)

120583+]119896 (z) [10] are generated by

119903

prod119894=1

(1 minus 119909119894119905)minus120572119894 =

infin

sum119899=0

g(1205721120572119903)119899

(1199091 119909

119903) 119905119899

(120572119894isin C (119894 = 1 2 119903) |119905| lt min 10038161003816100381610038161199091

1003816100381610038161003816minus1

10038161003816100381610038161199091199031003816100381610038161003816minus1

) (10)

We are thus led to the following result which provides aclass of bilateral generating functions for the Chan-Chyan-Srivastava polynomials and the family of multivariable poly-nomials given explicitly by (3)

Corollary 3 If Λ120583](z 119908) = suminfin

119896=0119886119896g (1205731 120573119903)

120583+]119896 (z)119908119896 119886119896

= 0120583 ] isin C z = (119911

1 119911

119903) then one has

infin

sum1198991 119899119903=0

[1198991119901]

sum119896=0

119886119896Θ(120572)

1198991minus1199011198961198992 119899119903

(x ym)


119903g (1205731 120573119903)

120583+]119896 (z) 1205781198961199051198991minus1199011198961

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

Λ120583] (z 120578)

(11)




119896= 1 120583 = 0 ] =

1 we find that

infin

sum1198991 119899119903=0

[1198991119901]

sum119896=0

Θ(120572)

1198991minus1199011198961198992 119899119903


119903g(1205731 120573119903)119896

(z)

times 1205781198961199051198991minus1199011198961

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572119903

prod119894=1

(1 minus 119911119894120578)

minus120573119894

10038161003816100381610038161205781003816100381610038161003816 lt min 10038161003816100381610038161199111

1003816100381610038161003816minus1

10038161003816100381610038161199111199031003816100381610038161003816minus1

1003816100381610038161003816100381610038161003816100381610038161003816

119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894)1003816100381610038161003816100381610038161003816100381610038161003816lt 1

(12)


1205931205821+12057811198961 120582119903+120578119903119896119903

(z) = Θ(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(u km) (13)

120582119894 120578

119894(119894 = 1 2 119903) isin N

0 u = (119906

1 119906

119903) k = (V

1



Corollary 5 If

Λnp120582120578120572120573120574m (x y u kw)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

Θ(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(x ym)

times Θ(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(u km) 1199081198961

1sdot sdot sdot 119908119896119903

119903

(14)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 then

we get

1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

Θ(120572)

1198991minus1198971 119899119903minus119897119903

(x ym)

times Θ(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(x ym)

timesΘ(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(u km)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Λnp120582120578120572120573120574m (x y u kw)

(15)




120583+]119896(y) and1205931205821+12057811198961 120582119903+120578119903119896119903

(y) y = (1199101 119910






Theorem 6 Let Ψ1198991119899119903


119866 (119898111990911199051minus 119910

111990511989811 119898

119903119909119903119905119903minus 119910

119903119905119898119903119903)

=infin

sum1198991 119899119903=0

Ψ1198991119899119903


119903

(16)


119899119894Ψ1198991119899119903

(x ym) = 119909119894

120597

120597119909119894

Ψ1198991119899119903

(x ym)

minus 119910119894

120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

(17)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

119899119894Ψ1198991119899119903

(x ym) = 119909119894

120597

120597119909119894

Ψ1198991119899119903

(x ym) (18)

for 119899119894= 0 1 119898

119894minus 2 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 Also one

finds that

119898119894119909119894

120597

120597119910119894

Ψ1198991119899119903

(x ym) minus 119898119894119910119894

120597

120597119910119894


(x ym)

= minus (119899119894minus 119898

119894+ 1)Ψ

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903(x ym)

(19)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Ψ1198991119899119903

(x ym) = 0 (20)

for 119899119894= 0 1 119898

119894minus 2 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894



119894and 119905


obtain thatinfin

sum1198991119899119903=0

119899119894Ψ1198991119899119903


119903

= (119909119894minus 119910

119894119905119898119894minus1119894

)infin

sum1198991119899119903=0

120597

120597119909119894

Ψ1198991119899119903


119903

= 119909119894

infin

sum1198991 119899119903=0

120597

120597119909119894

Ψ1198991119899119903


119903

minus 119910119894

infin

sum1198991 119899119894minus1119899119894+1119899119903=0

(119899119894ge119898119894minus1)

120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)


119903

(21)




119894and 119905


Theorem 7 If Ψ1198991119899119903


minus 119898119894

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1 119899119903

(x ym)

=120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894 119899119894+1119899119903

(x ym)

(22)

for 119899119894ge 119898

119894 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1119899119903

(x ym) = 0 (23)

for 119899119894= 1 2 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


and 119910119894 we have

minus 119898119894119905119894

infin

sum1198991 119899119903=0

120597

120597119910119894

Ψ1198991119899119903


119903

= 119905119898119894119894

infin

sum1198991 119899119903=0

120597

120597119909119894

Ψ1198991 119899119903


119903

(24)

which implies that

minus 119898119894

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1 119899119903

(x ym)

=120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894 119899119894+1119899119903

(x ym)

(25)

for 119899119894ge 119898

119894 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1119899119903

(x ym) = 0 (26)

for 119899119894= 1 2 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 Thus



119866 (119898111990911199051minus 119910

111990511989811 119898

119903119909119903119905119903minus 119910

119903119905119898119903119903)

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

(27)



119899119894Θ(120572)

1198991 119899119903

(x ym) = 119909119894

120597

120597119909119894

Θ(120572)

1198991119899119903

(x ym)

minus 119910119894

120597

120597119909119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(x ym)

(28)

119898119894119909119894

120597

120597119910119894

Θ(120572)

1198991 119899119903

(x ym) minus 119898119894119910119894

120597

120597119910119894

times Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(x ym)

= minus (119899119894minus 119898

119894+ 1)Θ(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

(29)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 Also for 119899

119894=

0 1 119898119894minus 2 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 we have

119899119894Θ(120572)

1198991 119899119903

(x ym) = 119909119894

120597

120597119909119894

Θ(120572)

1198991119899119903

(x ym) (30)

120597

120597119910119894

Θ(120572)

1198991119899119903

(x ym) = 0 (31)


119903

sum119894=1

119899119894Θ(120572)

1198991119899119903

(x ym)

=119903

sum119894=1

119909119894

120597

120597119909119894

Θ(120572)

1198991119899119903

(x ym)

minus119903

sum119894=1

119910119894

120597

120597119909119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(x ym)


119903

sum119894=1

119898119894119909119894

120597

120597119910119894

Θ(120572)

1198991 119899119903

(x ym)

minus119903

sum119894=1

119898119894119910119894

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

= minus119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

(32)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894



1198991119899119903

(x ym) are

minus 119898119894

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus1119899119894+1 119899119903

(x ym)

=120597

120597119909119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894 119899119894+1119899119903

(x ym)

(33)

for 119899119894ge 119898

119894 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(x ym) = 0 (34)

for 119899119894= 1 2 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


Θ(120572+120573)

1198991119899119903

(x ym) =1198991

sum1198961=0


sum119896119903=0

Θ(120572)

1198991minus1198961 119899119903minus119896119903

(x ym)

times Θ(120573)

1198961 119896119903

(x ym)

(35)


infin

sum1198991 119899119903=0

Θ(120572+120573)

1198991119899119903


119903

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus(120572+120573)

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572


sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120573

=infin

sum1198991 119899119903=0

Θ(120572)

1198991 119899119903


119903

timesinfin

sum1198961 119896119903=0

Θ(120573)

1198961 119896119903


119903

(36)

Replacing 119899119894by 119899

119894minus 119896



infin

sum1198991119899119903=0

1198991

sum1198961=0


sum119896119903=0

Θ(120572)

1198991minus1198961 119899119903minus119896119903

(x ym)

times Θ(120573)

1198961 119896119903


119903

(37)




1198991119899119903



1198991119899119903


Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119896119894119904119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2)


Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(]119894+ 119899

119894minus 119898

119894119896119894minus 2119904

119894)

119896119894119904119894(]

119894)119899119894minus119898119894119896119894minus119904119894+1

times(minus119910119894)119896119894119862]119894

119899119894minus119898119894119896119894minus2119904119894

(119898119894119909119894

2)


Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898i]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(minus119910119894)119896119894119867

119899119894minus119898119894119896119894minus2119904119894(119898

1198941199091198942)

119896119894119904119894 (119899

119894minus 119898

119894119896119894minus 2119904

119894)



Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

119899119894minus119898119894119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus119898119894119896119894

2119899119894minus119898119894119896119894(minus1)119904119894

119896119894 (119899

119894minus 119898

119894119896119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus119910119894)119896119894119871(120573119894)

119904119894

(119898119894119909119894

2)


(38)

Proof By (2) we get

infin

sum1198991119899119903=0

Θ(120572)

1198991 119899119903


119903

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

=infin

sum1198991 119899119903=0

infin

sum1198961 119896119903=0



1199031198991 sdot sdot sdot 119899



119903119909119903)1198991199031199101198961

1sdot sdot sdot 119910119896119903

119903

times 1199051198991+119898111989611

sdot sdot sdot 119905119899119903+119898119903119896119903119903

(39)


(119898119909)119899

119899=

[1198992]

sum119904=0

(2119899 minus 4119904 + 1) 119875119899minus2119904

(1198981199092)

119904(32)119899minus119904

(40)

we can write that

infin

sum1198991119899119903=0

Θ(120572)

1198991 119899119903


119903

=119903

prod119894=1

infin

sum119899119894=0

infin

sum119896119894=0

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119898119894119909119894

2)

times(minus119910119894)119896119894119905119899119894+119898119894119896119894

119894


(41)

Getting 119899119894minus 119898

119894119896119894instead of 119899


infin

sum1198991 119899119903=0

Θ(120572)

1198991119899119903


119903

=infin

sum1198991 119899119903=0

119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2) (minus119910

119894)119896119894119905119899119894

119894


=infin

sum1198991 119899119903=0

119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2) (minus119910

119894)119896119894


11990511989911sdot sdot sdot 119905119899119903

119903

(42)





(119898119909)119899

119899=

[1198992]

sum119896=0

(] + 119899 minus 2119896) 119862]119899minus2119896

(1198981199092)

119896(])119899+1minus119896

(119898119909)119899

119899=

[1198992]

sum119896=0

119867119899minus2119896

(1198981199092)

119896 (119899 minus 2119896)

(119898119909)119899

119899= 2119899

119899

sum119896=0

(minus1)119896(120572 + 1)119899119871(120572)119896

(1198981199092)

(119899 minus 119896)(120572 + 1)119896

(43)


1198991 119899119903



119899(x y m) and

Some Properties



properties

51TheCase of119898119894= 2 119910

119894= 1 119894 = 1 2 119903 in (2) This case


(1 minus 211990911199051+ 1199052


119903119905119903+ 1199052

119903)minus120572

=infin

sum1198991 119899119903=0

119862(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(44)



119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0


)


119903 (119899

1minus 2119896

1)


119903))

minus1

)


119903)119899119903minus2119896119903

(45)


1198991 119899119903

(1199091 119909



119894(119894 = 1 2 119903)

Thus 119862(120572)

1198991 119899119903

(1199091 119909


1+


1 119909


also yields

119862(120572)

1198991 119899119903

(1199091 119909

119903)

=21198991+sdotsdotsdot+119899119903(120572)

1198991+sdotsdotsdot+119899119903


119903

11990911989911sdot sdot sdot 119909119899119903

119903+ Π (119909

1 119909

119903)

(46)

whereΠ(1199091 119909



119903minus



1199091rarr minus119909

1and 119905

1rarr minus119905

1 we have

119862(120572)

1198991 119899119903

(minus1199091 119909

2 119909

119903) = (minus1)1198991119862(120572)

1198991 119899119903

(1199091 119909

119903)

(47)


119862(120572)

1198991 119899119903

(1199091 119909


119894 119909

119894+1 119909

119903)

= (minus1)119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

(48)


119894and 119905


119894 119894 = 1 2 119903 in (44)

we obtain

119862(120572)

1198991 119899119903



1198991 119899119903

(1199091 119909

119903)

(49)


1198991 119899119903

(1199091 119909

119903) one has

119862(120572)

21198991 2119899119903


1198991+sdotsdotsdot+119899119903


119903

(50)


119862(120572)

1198991 119899119903

(0 0 0) = 0 (51)


(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991119899119903=0

119862(120572)

1198991 119899119903

(0 0 0) 11990511989911sdot sdot sdot 119905119899119903

119903

(52)


(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991 119899119903=0



119903(120572)

1198991+sdotsdotsdot+119899119903119905211989911

sdot sdot sdot 1199052119899119903119903

(53)



desired


1198991 119899119903

(1199091 119909

119903)


119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

= 119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus119903

sum119894=1

120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

(54)

hold for 119899119894ge 1 119894 = 1 2 119903


1198991 119899119903

(1199091 119909


119862(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119862(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119862(120573)

1198961 119896119903

(1199091 119909

119903)

(55)



1198991 119899119903

(1199091 119909



119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(2119899119894minus 4119896

119894minus 4119904

119894+ 1)

119896119894119904119894(32)

119899119894minus2119896119894minus119904119894


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(]119894+ 119899

119894minus 2119896

119894minus 2119904

119894)

119896119894119904119894(]

119894)119899119894minus2119896119894minus119904119894+1


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0


(119909119894)

119896119894119904119894 (119899

119894minus 2119896

119894minus 2119904

119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

119899119894minus2119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus2119896119894

2119899119894minus2119896119894(minus1)119904119894

119896119894 (119899

119894minus 2119896

119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus1)119896119894119871(120573119894)119904119894

(119909119894)


(56)


1198991 119899119903

(1199091 119909

119903) given by (44)


1198991 119899119903

(1199091


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

times 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

(max 1

11990921

1

1199092119903

lt 1)

(57)

where 119865(119903)119861


119865(119903)119861

[1198861 119886

119903 1198871 119887

119903 119888 119909

1 119909

119903]

=infin

sum1198991119899119903=0

(1198861)1198991

sdot sdot sdot (119886119903)119899119903

(1198871)1198991

sdot sdot sdot (119887119903)119899119903

(119888)1198991+sdotsdotsdot+119899119903

11990911989911


119909119899119903119903

119899119903

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091199031003816100381610038161003816 lt 1)

(58)


(120572)119899minus119896

=(minus1)119896(120572)

119899

(1 minus 120572 minus 119899)119896

(minus119899)2119896

= 22119896(minus119899

2)119896

(minus119899 + 1

2)119896

(119899 minus 2119896) =119899

(minus119899)2119896

(59)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0

(((120572)1198991+sdotsdotsdot+119899119903


2)1198961

times (minus119899

1+ 1

2)1198961


2)119896119903

times(minus119899

119903+ 1

2)119896119903

)


1199031198991 sdot sdot sdot 119899

119903


119903)1198961+sdotsdotsdot+119896119903

)minus1

)


119903)119899119903minus2119896119903

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903


times[11989912]

sum1198961=0


sum119896119903=0

(((minus1198991

2)1198961

(minus119899

1+ 1

2)1198961


2)119896119903

(minus119899

119903+ 1

2)119896119903

)


119903)1198961+sdotsdotsdot+119896119903


119903)

minus1

)

times(1

11990921

)1198961

sdot sdot sdot (1

1199092119903

)119896119903

= 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

times(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

(60)

wheremax 111990921 11199092



52 The Case of 119909119894= 0 119910

119894= minus119909

119894 119894 = 1 2 119903 in (2) This

case yields

(1 minus 119909111990511989811

minus 119909211990511989822


=infin

sum1198991 119899119903=0

119880(120572)

1198991119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(61)



1198991119899119903

(1199091 119909

119903) satisfy

119898119894119909119894

120597

120597119909119894

119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

= (119899119894minus 119898

119894+ 1)119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119898119894119909119894

120597

120597119909119894

times 119880(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times 119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

(62)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


mula

119880(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119880(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119880(120573)

1198961 119896119903

(1199091 119909

119903)

(63)




119903119905119903)minus120572

=infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(64)


119866(120572)

1198991 119899119903

(1199091 119909

119903) =

(120572)1198991+sdotsdotsdot+119899119903


11990311990911989911sdot sdot sdot 119909119899119903

119903 (65)


1198991 119899119903

(1199091 119909

119903) satisfy

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903) = 119909

119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903) (66)

for 119894 = 1 2 119903 and119903

sum119894=1

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903)

(67)


1198991 119899119903

(1199091 119909


119866(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119866(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119866(120573)

1198961 119896119903

(1199091 119909

119903)

(68)



1198991 119899119903

(1199091



119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(]119894+ 119899

119894minus 2119904

119894)

119904119894(]

119894)119899119894minus119904119894+1

119862]119894119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

119867119899119894minus2119904119894

(1199091198942)

119904119894 (119899

119894minus 2119904

119894) (120572)

1198991+sdotsdotsdot+119899119903

119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

119899119894

sum119904119894=0

(120573119894+ 1)

119899119894

2119899119894(minus1)119904119894

(119899119894minus 119904

119894)(120573

119894+ 1)

119904119894

119871(120573119894)119904119894

(119909119894

2)


(69)


1198991 119899119903

(1199091 119909

119903)


1198991 119899119903

(1199091 119909

119903) the


infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(70)

for each 119894 = 1 2 119903 where


0= 1 (71)

and g (120572120573)


defined by [14]


=infin

sum119899=0

g (120572120573)

119899(119909 119910) 119905119899


)

(72)


120585119894


119894 centered 120585

119894= 119911

119894and 120585


0 lt 120576119894lt 1003816100381610038161003816119909119894

1003816100381610038161003816minus1

10038161003816100381610038161003816100381610038161003816100381610038161 minus

119903

sum119896=1

119909119896119911119896

1003816100381610038161003816100381610038161003816100381610038161003816 (73)


119863119898

119911119894(1 minus 119909


119903119911119903)minus120572

119911119896

119894

=

119898

2120587119894

∮C120585119894

120585119896

119894(1 minus 119909



119903119911119903)minus120572

(120585119894minus 119911

119894)119898+1

119889120585119894

=

119898

2120587119894


(120585119894+ 119911

119894)119896


119894(120585119894+ 119911


119903119911119903)minus120572

120585119898+1

119894

119889120585119894

=

119898

2120587119894

119911119896

119894(1 minus 119909


119903119911119903)minus120572


(1 + 120585119894119911

119894)119896

120585119898+1

119894

(1 minus

119909119894120585119894


119903119911119903

)

minus120572

119889120585119894

= 119898119911119896

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894


119903119911119903

minus

1

119911119894

)

= 119898119911119896minus119898

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894119911119894


119903119911119903

minus1)

(74)


119863119898

119911119894


119903119911119903)minus120572

119911119896119894

= 119863119898

119911119894

119911119896119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903

=infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903)119889119898

119889119911119898119894

11991111989911sdot sdot sdot 119911119899119894+119896

119894sdot sdot sdot 119911119899119903

119903

= 119911119896minus119898119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903

(75)


infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572

times g(120572minus119896)119898

(119909119894119911119894


119903119911119903

minus1)

(76)


[119899119894+ 119898]

119898= (119899

119894+ 119898) (119899


119894+ 1)

= (119899119894+ 1)

119898

(77)



infin

sum1198991 119899119903=0

(119899119894+ 1)

119898119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(78)



(1 minus 11990911199051minus 119909


119903119905119903119903)minus120572

=infin

sum1198991119899119903=0

119867(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(79)



1198991 119899119903

(1199091 119909

119903) satisfy

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

= (119899119894minus 119894 + 1)119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

119903

sum119894=1

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119894 + 1)

times 119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

(80)

for 119899119894ge 119894 minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


119867(120572+120573)

1198991119899119903

(1199091 119909

119903)

=1198991

sum1198961=0


sum119896119903=0

119867(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119867(120573)

1198961 119896119903

(1199091 119909

119903)

(81)


= 119894 119909119894

= 0 119910119894

= minus119909119894


1198991 119899119903

(1199091 119909

119903)

Remark 28 If

Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)1199081198961

1sdot sdot sdot 119908119896119903

119903

(82)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 and

120582 = (1205821 120582

119903) 120578 = (120578

1 120578

119903)w = (119908

1 119908

119903) then we

have1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572)

1198991minus1198971 119899119903minus119897119903

(1199091 119909

119903)

times 119867(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

(83)

References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896= 1 120583 = 0 ] =

1 we find that

infin

sum1198991 119899119903=0

[1198991119901]

sum119896=0

Θ(120572)

1198991minus1199011198961198992 119899119903


119903g(1205731 120573119903)119896

(z)

times 1205781198961199051198991minus1199011198961

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572119903

prod119894=1

(1 minus 119911119894120578)

minus120573119894

10038161003816100381610038161205781003816100381610038161003816 lt min 10038161003816100381610038161199111

1003816100381610038161003816minus1

10038161003816100381610038161199111199031003816100381610038161003816minus1

1003816100381610038161003816100381610038161003816100381610038161003816

119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894)1003816100381610038161003816100381610038161003816100381610038161003816lt 1

(12)


1205931205821+12057811198961 120582119903+120578119903119896119903

(z) = Θ(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(u km) (13)

120582119894 120578

119894(119894 = 1 2 119903) isin N

0 u = (119906

1 119906

119903) k = (V

1



Corollary 5 If

Λnp120582120578120572120573120574m (x y u kw)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

Θ(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(x ym)

times Θ(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(u km) 1199081198961

1sdot sdot sdot 119908119896119903

119903

(14)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 then

we get

1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

Θ(120572)

1198991minus1198971 119899119903minus119897119903

(x ym)

times Θ(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(x ym)

timesΘ(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(u km)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Λnp120582120578120572120573120574m (x y u kw)

(15)




120583+]119896(y) and1205931205821+12057811198961 120582119903+120578119903119896119903

(y) y = (1199101 119910






Theorem 6 Let Ψ1198991119899119903


119866 (119898111990911199051minus 119910

111990511989811 119898

119903119909119903119905119903minus 119910

119903119905119898119903119903)

=infin

sum1198991 119899119903=0

Ψ1198991119899119903


119903

(16)


119899119894Ψ1198991119899119903

(x ym) = 119909119894

120597

120597119909119894

Ψ1198991119899119903

(x ym)

minus 119910119894

120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

(17)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

119899119894Ψ1198991119899119903

(x ym) = 119909119894

120597

120597119909119894

Ψ1198991119899119903

(x ym) (18)

for 119899119894= 0 1 119898

119894minus 2 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 Also one

finds that

119898119894119909119894

120597

120597119910119894

Ψ1198991119899119903

(x ym) minus 119898119894119910119894

120597

120597119910119894


(x ym)

= minus (119899119894minus 119898

119894+ 1)Ψ

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903(x ym)

(19)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Ψ1198991119899119903

(x ym) = 0 (20)

for 119899119894= 0 1 119898

119894minus 2 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894



119894and 119905


obtain thatinfin

sum1198991119899119903=0

119899119894Ψ1198991119899119903


119903

= (119909119894minus 119910

119894119905119898119894minus1119894

)infin

sum1198991119899119903=0

120597

120597119909119894

Ψ1198991119899119903


119903

= 119909119894

infin

sum1198991 119899119903=0

120597

120597119909119894

Ψ1198991119899119903


119903

minus 119910119894

infin

sum1198991 119899119894minus1119899119894+1119899119903=0

(119899119894ge119898119894minus1)

120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)


119903

(21)




119894and 119905


Theorem 7 If Ψ1198991119899119903


minus 119898119894

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1 119899119903

(x ym)

=120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894 119899119894+1119899119903

(x ym)

(22)

for 119899119894ge 119898

119894 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1119899119903

(x ym) = 0 (23)

for 119899119894= 1 2 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


and 119910119894 we have

minus 119898119894119905119894

infin

sum1198991 119899119903=0

120597

120597119910119894

Ψ1198991119899119903


119903

= 119905119898119894119894

infin

sum1198991 119899119903=0

120597

120597119909119894

Ψ1198991 119899119903


119903

(24)

which implies that

minus 119898119894

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1 119899119903

(x ym)

=120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894 119899119894+1119899119903

(x ym)

(25)

for 119899119894ge 119898

119894 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1119899119903

(x ym) = 0 (26)

for 119899119894= 1 2 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 Thus



119866 (119898111990911199051minus 119910

111990511989811 119898

119903119909119903119905119903minus 119910

119903119905119898119903119903)

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

(27)



119899119894Θ(120572)

1198991 119899119903

(x ym) = 119909119894

120597

120597119909119894

Θ(120572)

1198991119899119903

(x ym)

minus 119910119894

120597

120597119909119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(x ym)

(28)

119898119894119909119894

120597

120597119910119894

Θ(120572)

1198991 119899119903

(x ym) minus 119898119894119910119894

120597

120597119910119894

times Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(x ym)

= minus (119899119894minus 119898

119894+ 1)Θ(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

(29)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 Also for 119899

119894=

0 1 119898119894minus 2 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 we have

119899119894Θ(120572)

1198991 119899119903

(x ym) = 119909119894

120597

120597119909119894

Θ(120572)

1198991119899119903

(x ym) (30)

120597

120597119910119894

Θ(120572)

1198991119899119903

(x ym) = 0 (31)


119903

sum119894=1

119899119894Θ(120572)

1198991119899119903

(x ym)

=119903

sum119894=1

119909119894

120597

120597119909119894

Θ(120572)

1198991119899119903

(x ym)

minus119903

sum119894=1

119910119894

120597

120597119909119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(x ym)


119903

sum119894=1

119898119894119909119894

120597

120597119910119894

Θ(120572)

1198991 119899119903

(x ym)

minus119903

sum119894=1

119898119894119910119894

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

= minus119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

(32)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894



1198991119899119903

(x ym) are

minus 119898119894

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus1119899119894+1 119899119903

(x ym)

=120597

120597119909119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894 119899119894+1119899119903

(x ym)

(33)

for 119899119894ge 119898

119894 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(x ym) = 0 (34)

for 119899119894= 1 2 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


Θ(120572+120573)

1198991119899119903

(x ym) =1198991

sum1198961=0


sum119896119903=0

Θ(120572)

1198991minus1198961 119899119903minus119896119903

(x ym)

times Θ(120573)

1198961 119896119903

(x ym)

(35)


infin

sum1198991 119899119903=0

Θ(120572+120573)

1198991119899119903


119903

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus(120572+120573)

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572


sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120573

=infin

sum1198991 119899119903=0

Θ(120572)

1198991 119899119903


119903

timesinfin

sum1198961 119896119903=0

Θ(120573)

1198961 119896119903


119903

(36)

Replacing 119899119894by 119899

119894minus 119896



infin

sum1198991119899119903=0

1198991

sum1198961=0


sum119896119903=0

Θ(120572)

1198991minus1198961 119899119903minus119896119903

(x ym)

times Θ(120573)

1198961 119896119903


119903

(37)




1198991119899119903



1198991119899119903


Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119896119894119904119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2)


Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(]119894+ 119899

119894minus 119898

119894119896119894minus 2119904

119894)

119896119894119904119894(]

119894)119899119894minus119898119894119896119894minus119904119894+1

times(minus119910119894)119896119894119862]119894

119899119894minus119898119894119896119894minus2119904119894

(119898119894119909119894

2)


Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898i]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(minus119910119894)119896119894119867

119899119894minus119898119894119896119894minus2119904119894(119898

1198941199091198942)

119896119894119904119894 (119899

119894minus 119898

119894119896119894minus 2119904

119894)



Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

119899119894minus119898119894119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus119898119894119896119894

2119899119894minus119898119894119896119894(minus1)119904119894

119896119894 (119899

119894minus 119898

119894119896119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus119910119894)119896119894119871(120573119894)

119904119894

(119898119894119909119894

2)


(38)

Proof By (2) we get

infin

sum1198991119899119903=0

Θ(120572)

1198991 119899119903


119903

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

=infin

sum1198991 119899119903=0

infin

sum1198961 119896119903=0



1199031198991 sdot sdot sdot 119899



119903119909119903)1198991199031199101198961

1sdot sdot sdot 119910119896119903

119903

times 1199051198991+119898111989611

sdot sdot sdot 119905119899119903+119898119903119896119903119903

(39)


(119898119909)119899

119899=

[1198992]

sum119904=0

(2119899 minus 4119904 + 1) 119875119899minus2119904

(1198981199092)

119904(32)119899minus119904

(40)

we can write that

infin

sum1198991119899119903=0

Θ(120572)

1198991 119899119903


119903

=119903

prod119894=1

infin

sum119899119894=0

infin

sum119896119894=0

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119898119894119909119894

2)

times(minus119910119894)119896119894119905119899119894+119898119894119896119894

119894


(41)

Getting 119899119894minus 119898

119894119896119894instead of 119899


infin

sum1198991 119899119903=0

Θ(120572)

1198991119899119903


119903

=infin

sum1198991 119899119903=0

119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2) (minus119910

119894)119896119894119905119899119894

119894


=infin

sum1198991 119899119903=0

119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2) (minus119910

119894)119896119894


11990511989911sdot sdot sdot 119905119899119903

119903

(42)





(119898119909)119899

119899=

[1198992]

sum119896=0

(] + 119899 minus 2119896) 119862]119899minus2119896

(1198981199092)

119896(])119899+1minus119896

(119898119909)119899

119899=

[1198992]

sum119896=0

119867119899minus2119896

(1198981199092)

119896 (119899 minus 2119896)

(119898119909)119899

119899= 2119899

119899

sum119896=0

(minus1)119896(120572 + 1)119899119871(120572)119896

(1198981199092)

(119899 minus 119896)(120572 + 1)119896

(43)


1198991 119899119903



119899(x y m) and

Some Properties



properties

51TheCase of119898119894= 2 119910

119894= 1 119894 = 1 2 119903 in (2) This case


(1 minus 211990911199051+ 1199052


119903119905119903+ 1199052

119903)minus120572

=infin

sum1198991 119899119903=0

119862(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(44)



119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0


)


119903 (119899

1minus 2119896

1)


119903))

minus1

)


119903)119899119903minus2119896119903

(45)


1198991 119899119903

(1199091 119909



119894(119894 = 1 2 119903)

Thus 119862(120572)

1198991 119899119903

(1199091 119909


1+


1 119909


also yields

119862(120572)

1198991 119899119903

(1199091 119909

119903)

=21198991+sdotsdotsdot+119899119903(120572)

1198991+sdotsdotsdot+119899119903


119903

11990911989911sdot sdot sdot 119909119899119903

119903+ Π (119909

1 119909

119903)

(46)

whereΠ(1199091 119909



119903minus



1199091rarr minus119909

1and 119905

1rarr minus119905

1 we have

119862(120572)

1198991 119899119903

(minus1199091 119909

2 119909

119903) = (minus1)1198991119862(120572)

1198991 119899119903

(1199091 119909

119903)

(47)


119862(120572)

1198991 119899119903

(1199091 119909


119894 119909

119894+1 119909

119903)

= (minus1)119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

(48)


119894and 119905


119894 119894 = 1 2 119903 in (44)

we obtain

119862(120572)

1198991 119899119903



1198991 119899119903

(1199091 119909

119903)

(49)


1198991 119899119903

(1199091 119909

119903) one has

119862(120572)

21198991 2119899119903


1198991+sdotsdotsdot+119899119903


119903

(50)


119862(120572)

1198991 119899119903

(0 0 0) = 0 (51)


(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991119899119903=0

119862(120572)

1198991 119899119903

(0 0 0) 11990511989911sdot sdot sdot 119905119899119903

119903

(52)


(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991 119899119903=0



119903(120572)

1198991+sdotsdotsdot+119899119903119905211989911

sdot sdot sdot 1199052119899119903119903

(53)



desired


1198991 119899119903

(1199091 119909

119903)


119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

= 119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus119903

sum119894=1

120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

(54)

hold for 119899119894ge 1 119894 = 1 2 119903


1198991 119899119903

(1199091 119909


119862(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119862(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119862(120573)

1198961 119896119903

(1199091 119909

119903)

(55)



1198991 119899119903

(1199091 119909



119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(2119899119894minus 4119896

119894minus 4119904

119894+ 1)

119896119894119904119894(32)

119899119894minus2119896119894minus119904119894


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(]119894+ 119899

119894minus 2119896

119894minus 2119904

119894)

119896119894119904119894(]

119894)119899119894minus2119896119894minus119904119894+1


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0


(119909119894)

119896119894119904119894 (119899

119894minus 2119896

119894minus 2119904

119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

119899119894minus2119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus2119896119894

2119899119894minus2119896119894(minus1)119904119894

119896119894 (119899

119894minus 2119896

119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus1)119896119894119871(120573119894)119904119894

(119909119894)


(56)


1198991 119899119903

(1199091 119909

119903) given by (44)


1198991 119899119903

(1199091


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

times 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

(max 1

11990921

1

1199092119903

lt 1)

(57)

where 119865(119903)119861


119865(119903)119861

[1198861 119886

119903 1198871 119887

119903 119888 119909

1 119909

119903]

=infin

sum1198991119899119903=0

(1198861)1198991

sdot sdot sdot (119886119903)119899119903

(1198871)1198991

sdot sdot sdot (119887119903)119899119903

(119888)1198991+sdotsdotsdot+119899119903

11990911989911


119909119899119903119903

119899119903

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091199031003816100381610038161003816 lt 1)

(58)


(120572)119899minus119896

=(minus1)119896(120572)

119899

(1 minus 120572 minus 119899)119896

(minus119899)2119896

= 22119896(minus119899

2)119896

(minus119899 + 1

2)119896

(119899 minus 2119896) =119899

(minus119899)2119896

(59)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0

(((120572)1198991+sdotsdotsdot+119899119903


2)1198961

times (minus119899

1+ 1

2)1198961


2)119896119903

times(minus119899

119903+ 1

2)119896119903

)


1199031198991 sdot sdot sdot 119899

119903


119903)1198961+sdotsdotsdot+119896119903

)minus1

)


119903)119899119903minus2119896119903

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903


times[11989912]

sum1198961=0


sum119896119903=0

(((minus1198991

2)1198961

(minus119899

1+ 1

2)1198961


2)119896119903

(minus119899

119903+ 1

2)119896119903

)


119903)1198961+sdotsdotsdot+119896119903


119903)

minus1

)

times(1

11990921

)1198961

sdot sdot sdot (1

1199092119903

)119896119903

= 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

times(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

(60)

wheremax 111990921 11199092



52 The Case of 119909119894= 0 119910

119894= minus119909

119894 119894 = 1 2 119903 in (2) This

case yields

(1 minus 119909111990511989811

minus 119909211990511989822


=infin

sum1198991 119899119903=0

119880(120572)

1198991119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(61)



1198991119899119903

(1199091 119909

119903) satisfy

119898119894119909119894

120597

120597119909119894

119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

= (119899119894minus 119898

119894+ 1)119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119898119894119909119894

120597

120597119909119894

times 119880(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times 119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

(62)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


mula

119880(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119880(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119880(120573)

1198961 119896119903

(1199091 119909

119903)

(63)




119903119905119903)minus120572

=infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(64)


119866(120572)

1198991 119899119903

(1199091 119909

119903) =

(120572)1198991+sdotsdotsdot+119899119903


11990311990911989911sdot sdot sdot 119909119899119903

119903 (65)


1198991 119899119903

(1199091 119909

119903) satisfy

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903) = 119909

119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903) (66)

for 119894 = 1 2 119903 and119903

sum119894=1

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903)

(67)


1198991 119899119903

(1199091 119909


119866(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119866(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119866(120573)

1198961 119896119903

(1199091 119909

119903)

(68)



1198991 119899119903

(1199091



119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(]119894+ 119899

119894minus 2119904

119894)

119904119894(]

119894)119899119894minus119904119894+1

119862]119894119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

119867119899119894minus2119904119894

(1199091198942)

119904119894 (119899

119894minus 2119904

119894) (120572)

1198991+sdotsdotsdot+119899119903

119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

119899119894

sum119904119894=0

(120573119894+ 1)

119899119894

2119899119894(minus1)119904119894

(119899119894minus 119904

119894)(120573

119894+ 1)

119904119894

119871(120573119894)119904119894

(119909119894

2)


(69)


1198991 119899119903

(1199091 119909

119903)


1198991 119899119903

(1199091 119909

119903) the


infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(70)

for each 119894 = 1 2 119903 where


0= 1 (71)

and g (120572120573)


defined by [14]


=infin

sum119899=0

g (120572120573)

119899(119909 119910) 119905119899


)

(72)


120585119894


119894 centered 120585

119894= 119911

119894and 120585


0 lt 120576119894lt 1003816100381610038161003816119909119894

1003816100381610038161003816minus1

10038161003816100381610038161003816100381610038161003816100381610038161 minus

119903

sum119896=1

119909119896119911119896

1003816100381610038161003816100381610038161003816100381610038161003816 (73)


119863119898

119911119894(1 minus 119909


119903119911119903)minus120572

119911119896

119894

=

119898

2120587119894

∮C120585119894

120585119896

119894(1 minus 119909



119903119911119903)minus120572

(120585119894minus 119911

119894)119898+1

119889120585119894

=

119898

2120587119894


(120585119894+ 119911

119894)119896


119894(120585119894+ 119911


119903119911119903)minus120572

120585119898+1

119894

119889120585119894

=

119898

2120587119894

119911119896

119894(1 minus 119909


119903119911119903)minus120572


(1 + 120585119894119911

119894)119896

120585119898+1

119894

(1 minus

119909119894120585119894


119903119911119903

)

minus120572

119889120585119894

= 119898119911119896

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894


119903119911119903

minus

1

119911119894

)

= 119898119911119896minus119898

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894119911119894


119903119911119903

minus1)

(74)


119863119898

119911119894


119903119911119903)minus120572

119911119896119894

= 119863119898

119911119894

119911119896119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903

=infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903)119889119898

119889119911119898119894

11991111989911sdot sdot sdot 119911119899119894+119896

119894sdot sdot sdot 119911119899119903

119903

= 119911119896minus119898119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903

(75)


infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572

times g(120572minus119896)119898

(119909119894119911119894


119903119911119903

minus1)

(76)


[119899119894+ 119898]

119898= (119899

119894+ 119898) (119899


119894+ 1)

= (119899119894+ 1)

119898

(77)



infin

sum1198991 119899119903=0

(119899119894+ 1)

119898119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(78)



(1 minus 11990911199051minus 119909


119903119905119903119903)minus120572

=infin

sum1198991119899119903=0

119867(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(79)



1198991 119899119903

(1199091 119909

119903) satisfy

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

= (119899119894minus 119894 + 1)119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

119903

sum119894=1

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119894 + 1)

times 119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

(80)

for 119899119894ge 119894 minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


119867(120572+120573)

1198991119899119903

(1199091 119909

119903)

=1198991

sum1198961=0


sum119896119903=0

119867(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119867(120573)

1198961 119896119903

(1199091 119909

119903)

(81)


= 119894 119909119894

= 0 119910119894

= minus119909119894


1198991 119899119903

(1199091 119909

119903)

Remark 28 If

Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)1199081198961

1sdot sdot sdot 119908119896119903

119903

(82)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 and

120582 = (1205821 120582

119903) 120578 = (120578

1 120578

119903)w = (119908

1 119908

119903) then we

have1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572)

1198991minus1198971 119899119903minus119897119903

(1199091 119909

119903)

times 119867(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

(83)

References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894and 119905


obtain thatinfin

sum1198991119899119903=0

119899119894Ψ1198991119899119903


119903

= (119909119894minus 119910

119894119905119898119894minus1119894

)infin

sum1198991119899119903=0

120597

120597119909119894

Ψ1198991119899119903


119903

= 119909119894

infin

sum1198991 119899119903=0

120597

120597119909119894

Ψ1198991119899119903


119903

minus 119910119894

infin

sum1198991 119899119894minus1119899119894+1119899119903=0

(119899119894ge119898119894minus1)

120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)


119903

(21)




119894and 119905


Theorem 7 If Ψ1198991119899119903


minus 119898119894

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1 119899119903

(x ym)

=120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894 119899119894+1119899119903

(x ym)

(22)

for 119899119894ge 119898

119894 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1119899119903

(x ym) = 0 (23)

for 119899119894= 1 2 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


and 119910119894 we have

minus 119898119894119905119894

infin

sum1198991 119899119903=0

120597

120597119910119894

Ψ1198991119899119903


119903

= 119905119898119894119894

infin

sum1198991 119899119903=0

120597

120597119909119894

Ψ1198991 119899119903


119903

(24)

which implies that

minus 119898119894

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1 119899119903

(x ym)

=120597

120597119909119894

Ψ1198991119899119894minus1119899119894minus119898119894 119899119894+1119899119903

(x ym)

(25)

for 119899119894ge 119898

119894 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Ψ1198991119899119894minus1119899119894minus1119899119894+1119899119903

(x ym) = 0 (26)

for 119899119894= 1 2 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 Thus



119866 (119898111990911199051minus 119910

111990511989811 119898

119903119909119903119905119903minus 119910

119903119905119898119903119903)

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

(27)



119899119894Θ(120572)

1198991 119899119903

(x ym) = 119909119894

120597

120597119909119894

Θ(120572)

1198991119899119903

(x ym)

minus 119910119894

120597

120597119909119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(x ym)

(28)

119898119894119909119894

120597

120597119910119894

Θ(120572)

1198991 119899119903

(x ym) minus 119898119894119910119894

120597

120597119910119894

times Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(x ym)

= minus (119899119894minus 119898

119894+ 1)Θ(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

(29)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 Also for 119899

119894=

0 1 119898119894minus 2 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 we have

119899119894Θ(120572)

1198991 119899119903

(x ym) = 119909119894

120597

120597119909119894

Θ(120572)

1198991119899119903

(x ym) (30)

120597

120597119910119894

Θ(120572)

1198991119899119903

(x ym) = 0 (31)


119903

sum119894=1

119899119894Θ(120572)

1198991119899119903

(x ym)

=119903

sum119894=1

119909119894

120597

120597119909119894

Θ(120572)

1198991119899119903

(x ym)

minus119903

sum119894=1

119910119894

120597

120597119909119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(x ym)


119903

sum119894=1

119898119894119909119894

120597

120597119910119894

Θ(120572)

1198991 119899119903

(x ym)

minus119903

sum119894=1

119898119894119910119894

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

= minus119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

(32)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894



1198991119899119903

(x ym) are

minus 119898119894

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus1119899119894+1 119899119903

(x ym)

=120597

120597119909119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894 119899119894+1119899119903

(x ym)

(33)

for 119899119894ge 119898

119894 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(x ym) = 0 (34)

for 119899119894= 1 2 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


Θ(120572+120573)

1198991119899119903

(x ym) =1198991

sum1198961=0


sum119896119903=0

Θ(120572)

1198991minus1198961 119899119903minus119896119903

(x ym)

times Θ(120573)

1198961 119896119903

(x ym)

(35)


infin

sum1198991 119899119903=0

Θ(120572+120573)

1198991119899119903


119903

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus(120572+120573)

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572


sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120573

=infin

sum1198991 119899119903=0

Θ(120572)

1198991 119899119903


119903

timesinfin

sum1198961 119896119903=0

Θ(120573)

1198961 119896119903


119903

(36)

Replacing 119899119894by 119899

119894minus 119896



infin

sum1198991119899119903=0

1198991

sum1198961=0


sum119896119903=0

Θ(120572)

1198991minus1198961 119899119903minus119896119903

(x ym)

times Θ(120573)

1198961 119896119903


119903

(37)




1198991119899119903



1198991119899119903


Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119896119894119904119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2)


Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(]119894+ 119899

119894minus 119898

119894119896119894minus 2119904

119894)

119896119894119904119894(]

119894)119899119894minus119898119894119896119894minus119904119894+1

times(minus119910119894)119896119894119862]119894

119899119894minus119898119894119896119894minus2119904119894

(119898119894119909119894

2)


Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898i]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(minus119910119894)119896119894119867

119899119894minus119898119894119896119894minus2119904119894(119898

1198941199091198942)

119896119894119904119894 (119899

119894minus 119898

119894119896119894minus 2119904

119894)



Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

119899119894minus119898119894119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus119898119894119896119894

2119899119894minus119898119894119896119894(minus1)119904119894

119896119894 (119899

119894minus 119898

119894119896119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus119910119894)119896119894119871(120573119894)

119904119894

(119898119894119909119894

2)


(38)

Proof By (2) we get

infin

sum1198991119899119903=0

Θ(120572)

1198991 119899119903


119903

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

=infin

sum1198991 119899119903=0

infin

sum1198961 119896119903=0



1199031198991 sdot sdot sdot 119899



119903119909119903)1198991199031199101198961

1sdot sdot sdot 119910119896119903

119903

times 1199051198991+119898111989611

sdot sdot sdot 119905119899119903+119898119903119896119903119903

(39)


(119898119909)119899

119899=

[1198992]

sum119904=0

(2119899 minus 4119904 + 1) 119875119899minus2119904

(1198981199092)

119904(32)119899minus119904

(40)

we can write that

infin

sum1198991119899119903=0

Θ(120572)

1198991 119899119903


119903

=119903

prod119894=1

infin

sum119899119894=0

infin

sum119896119894=0

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119898119894119909119894

2)

times(minus119910119894)119896119894119905119899119894+119898119894119896119894

119894


(41)

Getting 119899119894minus 119898

119894119896119894instead of 119899


infin

sum1198991 119899119903=0

Θ(120572)

1198991119899119903


119903

=infin

sum1198991 119899119903=0

119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2) (minus119910

119894)119896119894119905119899119894

119894


=infin

sum1198991 119899119903=0

119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2) (minus119910

119894)119896119894


11990511989911sdot sdot sdot 119905119899119903

119903

(42)





(119898119909)119899

119899=

[1198992]

sum119896=0

(] + 119899 minus 2119896) 119862]119899minus2119896

(1198981199092)

119896(])119899+1minus119896

(119898119909)119899

119899=

[1198992]

sum119896=0

119867119899minus2119896

(1198981199092)

119896 (119899 minus 2119896)

(119898119909)119899

119899= 2119899

119899

sum119896=0

(minus1)119896(120572 + 1)119899119871(120572)119896

(1198981199092)

(119899 minus 119896)(120572 + 1)119896

(43)


1198991 119899119903



119899(x y m) and

Some Properties



properties

51TheCase of119898119894= 2 119910

119894= 1 119894 = 1 2 119903 in (2) This case


(1 minus 211990911199051+ 1199052


119903119905119903+ 1199052

119903)minus120572

=infin

sum1198991 119899119903=0

119862(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(44)



119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0


)


119903 (119899

1minus 2119896

1)


119903))

minus1

)


119903)119899119903minus2119896119903

(45)


1198991 119899119903

(1199091 119909



119894(119894 = 1 2 119903)

Thus 119862(120572)

1198991 119899119903

(1199091 119909


1+


1 119909


also yields

119862(120572)

1198991 119899119903

(1199091 119909

119903)

=21198991+sdotsdotsdot+119899119903(120572)

1198991+sdotsdotsdot+119899119903


119903

11990911989911sdot sdot sdot 119909119899119903

119903+ Π (119909

1 119909

119903)

(46)

whereΠ(1199091 119909



119903minus



1199091rarr minus119909

1and 119905

1rarr minus119905

1 we have

119862(120572)

1198991 119899119903

(minus1199091 119909

2 119909

119903) = (minus1)1198991119862(120572)

1198991 119899119903

(1199091 119909

119903)

(47)


119862(120572)

1198991 119899119903

(1199091 119909


119894 119909

119894+1 119909

119903)

= (minus1)119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

(48)


119894and 119905


119894 119894 = 1 2 119903 in (44)

we obtain

119862(120572)

1198991 119899119903



1198991 119899119903

(1199091 119909

119903)

(49)


1198991 119899119903

(1199091 119909

119903) one has

119862(120572)

21198991 2119899119903


1198991+sdotsdotsdot+119899119903


119903

(50)


119862(120572)

1198991 119899119903

(0 0 0) = 0 (51)


(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991119899119903=0

119862(120572)

1198991 119899119903

(0 0 0) 11990511989911sdot sdot sdot 119905119899119903

119903

(52)


(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991 119899119903=0



119903(120572)

1198991+sdotsdotsdot+119899119903119905211989911

sdot sdot sdot 1199052119899119903119903

(53)



desired


1198991 119899119903

(1199091 119909

119903)


119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

= 119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus119903

sum119894=1

120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

(54)

hold for 119899119894ge 1 119894 = 1 2 119903


1198991 119899119903

(1199091 119909


119862(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119862(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119862(120573)

1198961 119896119903

(1199091 119909

119903)

(55)



1198991 119899119903

(1199091 119909



119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(2119899119894minus 4119896

119894minus 4119904

119894+ 1)

119896119894119904119894(32)

119899119894minus2119896119894minus119904119894


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(]119894+ 119899

119894minus 2119896

119894minus 2119904

119894)

119896119894119904119894(]

119894)119899119894minus2119896119894minus119904119894+1


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0


(119909119894)

119896119894119904119894 (119899

119894minus 2119896

119894minus 2119904

119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

119899119894minus2119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus2119896119894

2119899119894minus2119896119894(minus1)119904119894

119896119894 (119899

119894minus 2119896

119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus1)119896119894119871(120573119894)119904119894

(119909119894)


(56)


1198991 119899119903

(1199091 119909

119903) given by (44)


1198991 119899119903

(1199091


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

times 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

(max 1

11990921

1

1199092119903

lt 1)

(57)

where 119865(119903)119861


119865(119903)119861

[1198861 119886

119903 1198871 119887

119903 119888 119909

1 119909

119903]

=infin

sum1198991119899119903=0

(1198861)1198991

sdot sdot sdot (119886119903)119899119903

(1198871)1198991

sdot sdot sdot (119887119903)119899119903

(119888)1198991+sdotsdotsdot+119899119903

11990911989911


119909119899119903119903

119899119903

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091199031003816100381610038161003816 lt 1)

(58)


(120572)119899minus119896

=(minus1)119896(120572)

119899

(1 minus 120572 minus 119899)119896

(minus119899)2119896

= 22119896(minus119899

2)119896

(minus119899 + 1

2)119896

(119899 minus 2119896) =119899

(minus119899)2119896

(59)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0

(((120572)1198991+sdotsdotsdot+119899119903


2)1198961

times (minus119899

1+ 1

2)1198961


2)119896119903

times(minus119899

119903+ 1

2)119896119903

)


1199031198991 sdot sdot sdot 119899

119903


119903)1198961+sdotsdotsdot+119896119903

)minus1

)


119903)119899119903minus2119896119903

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903


times[11989912]

sum1198961=0


sum119896119903=0

(((minus1198991

2)1198961

(minus119899

1+ 1

2)1198961


2)119896119903

(minus119899

119903+ 1

2)119896119903

)


119903)1198961+sdotsdotsdot+119896119903


119903)

minus1

)

times(1

11990921

)1198961

sdot sdot sdot (1

1199092119903

)119896119903

= 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

times(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

(60)

wheremax 111990921 11199092



52 The Case of 119909119894= 0 119910

119894= minus119909

119894 119894 = 1 2 119903 in (2) This

case yields

(1 minus 119909111990511989811

minus 119909211990511989822


=infin

sum1198991 119899119903=0

119880(120572)

1198991119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(61)



1198991119899119903

(1199091 119909

119903) satisfy

119898119894119909119894

120597

120597119909119894

119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

= (119899119894minus 119898

119894+ 1)119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119898119894119909119894

120597

120597119909119894

times 119880(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times 119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

(62)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


mula

119880(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119880(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119880(120573)

1198961 119896119903

(1199091 119909

119903)

(63)




119903119905119903)minus120572

=infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(64)


119866(120572)

1198991 119899119903

(1199091 119909

119903) =

(120572)1198991+sdotsdotsdot+119899119903


11990311990911989911sdot sdot sdot 119909119899119903

119903 (65)


1198991 119899119903

(1199091 119909

119903) satisfy

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903) = 119909

119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903) (66)

for 119894 = 1 2 119903 and119903

sum119894=1

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903)

(67)


1198991 119899119903

(1199091 119909


119866(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119866(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119866(120573)

1198961 119896119903

(1199091 119909

119903)

(68)



1198991 119899119903

(1199091



119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(]119894+ 119899

119894minus 2119904

119894)

119904119894(]

119894)119899119894minus119904119894+1

119862]119894119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

119867119899119894minus2119904119894

(1199091198942)

119904119894 (119899

119894minus 2119904

119894) (120572)

1198991+sdotsdotsdot+119899119903

119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

119899119894

sum119904119894=0

(120573119894+ 1)

119899119894

2119899119894(minus1)119904119894

(119899119894minus 119904

119894)(120573

119894+ 1)

119904119894

119871(120573119894)119904119894

(119909119894

2)


(69)


1198991 119899119903

(1199091 119909

119903)


1198991 119899119903

(1199091 119909

119903) the


infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(70)

for each 119894 = 1 2 119903 where


0= 1 (71)

and g (120572120573)


defined by [14]


=infin

sum119899=0

g (120572120573)

119899(119909 119910) 119905119899


)

(72)


120585119894


119894 centered 120585

119894= 119911

119894and 120585


0 lt 120576119894lt 1003816100381610038161003816119909119894

1003816100381610038161003816minus1

10038161003816100381610038161003816100381610038161003816100381610038161 minus

119903

sum119896=1

119909119896119911119896

1003816100381610038161003816100381610038161003816100381610038161003816 (73)


119863119898

119911119894(1 minus 119909


119903119911119903)minus120572

119911119896

119894

=

119898

2120587119894

∮C120585119894

120585119896

119894(1 minus 119909



119903119911119903)minus120572

(120585119894minus 119911

119894)119898+1

119889120585119894

=

119898

2120587119894


(120585119894+ 119911

119894)119896


119894(120585119894+ 119911


119903119911119903)minus120572

120585119898+1

119894

119889120585119894

=

119898

2120587119894

119911119896

119894(1 minus 119909


119903119911119903)minus120572


(1 + 120585119894119911

119894)119896

120585119898+1

119894

(1 minus

119909119894120585119894


119903119911119903

)

minus120572

119889120585119894

= 119898119911119896

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894


119903119911119903

minus

1

119911119894

)

= 119898119911119896minus119898

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894119911119894


119903119911119903

minus1)

(74)


119863119898

119911119894


119903119911119903)minus120572

119911119896119894

= 119863119898

119911119894

119911119896119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903

=infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903)119889119898

119889119911119898119894

11991111989911sdot sdot sdot 119911119899119894+119896

119894sdot sdot sdot 119911119899119903

119903

= 119911119896minus119898119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903

(75)


infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572

times g(120572minus119896)119898

(119909119894119911119894


119903119911119903

minus1)

(76)


[119899119894+ 119898]

119898= (119899

119894+ 119898) (119899


119894+ 1)

= (119899119894+ 1)

119898

(77)



infin

sum1198991 119899119903=0

(119899119894+ 1)

119898119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(78)



(1 minus 11990911199051minus 119909


119903119905119903119903)minus120572

=infin

sum1198991119899119903=0

119867(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(79)



1198991 119899119903

(1199091 119909

119903) satisfy

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

= (119899119894minus 119894 + 1)119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

119903

sum119894=1

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119894 + 1)

times 119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

(80)

for 119899119894ge 119894 minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


119867(120572+120573)

1198991119899119903

(1199091 119909

119903)

=1198991

sum1198961=0


sum119896119903=0

119867(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119867(120573)

1198961 119896119903

(1199091 119909

119903)

(81)


= 119894 119909119894

= 0 119910119894

= minus119909119894


1198991 119899119903

(1199091 119909

119903)

Remark 28 If

Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)1199081198961

1sdot sdot sdot 119908119896119903

119903

(82)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 and

120582 = (1205821 120582

119903) 120578 = (120578

1 120578

119903)w = (119908

1 119908

119903) then we

have1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572)

1198991minus1198971 119899119903minus119897119903

(1199091 119909

119903)

times 119867(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

(83)

References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119903

sum119894=1

119898119894119909119894

120597

120597119910119894

Θ(120572)

1198991 119899119903

(x ym)

minus119903

sum119894=1

119898119894119910119894

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

= minus119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times Θ(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(x ym)

(32)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894



1198991119899119903

(x ym) are

minus 119898119894

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus1119899119894+1 119899119903

(x ym)

=120597

120597119909119894

Θ(120572)

1198991 119899119894minus1119899119894minus119898119894 119899119894+1119899119903

(x ym)

(33)

for 119899119894ge 119898

119894 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894 and

120597

120597119910119894

Θ(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(x ym) = 0 (34)

for 119899119894= 1 2 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


Θ(120572+120573)

1198991119899119903

(x ym) =1198991

sum1198961=0


sum119896119903=0

Θ(120572)

1198991minus1198961 119899119903minus119896119903

(x ym)

times Θ(120573)

1198961 119896119903

(x ym)

(35)


infin

sum1198991 119899119903=0

Θ(120572+120573)

1198991119899119903


119903

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus(120572+120573)

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572


sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120573

=infin

sum1198991 119899119903=0

Θ(120572)

1198991 119899119903


119903

timesinfin

sum1198961 119896119903=0

Θ(120573)

1198961 119896119903


119903

(36)

Replacing 119899119894by 119899

119894minus 119896



infin

sum1198991119899119903=0

1198991

sum1198961=0


sum119896119903=0

Θ(120572)

1198991minus1198961 119899119903minus119896119903

(x ym)

times Θ(120573)

1198961 119896119903


119903

(37)




1198991119899119903



1198991119899119903


Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119896119894119904119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2)


Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(]119894+ 119899

119894minus 119898

119894119896119894minus 2119904

119894)

119896119894119904119894(]

119894)119899119894minus119898119894119896119894minus119904119894+1

times(minus119910119894)119896119894119862]119894

119899119894minus119898119894119896119894minus2119904119894

(119898119894119909119894

2)


Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898i]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(minus119910119894)119896119894119867

119899119894minus119898119894119896119894minus2119904119894(119898

1198941199091198942)

119896119894119904119894 (119899

119894minus 119898

119894119896119894minus 2119904

119894)



Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

119899119894minus119898119894119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus119898119894119896119894

2119899119894minus119898119894119896119894(minus1)119904119894

119896119894 (119899

119894minus 119898

119894119896119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus119910119894)119896119894119871(120573119894)

119904119894

(119898119894119909119894

2)


(38)

Proof By (2) we get

infin

sum1198991119899119903=0

Θ(120572)

1198991 119899119903


119903

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

=infin

sum1198991 119899119903=0

infin

sum1198961 119896119903=0



1199031198991 sdot sdot sdot 119899



119903119909119903)1198991199031199101198961

1sdot sdot sdot 119910119896119903

119903

times 1199051198991+119898111989611

sdot sdot sdot 119905119899119903+119898119903119896119903119903

(39)


(119898119909)119899

119899=

[1198992]

sum119904=0

(2119899 minus 4119904 + 1) 119875119899minus2119904

(1198981199092)

119904(32)119899minus119904

(40)

we can write that

infin

sum1198991119899119903=0

Θ(120572)

1198991 119899119903


119903

=119903

prod119894=1

infin

sum119899119894=0

infin

sum119896119894=0

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119898119894119909119894

2)

times(minus119910119894)119896119894119905119899119894+119898119894119896119894

119894


(41)

Getting 119899119894minus 119898

119894119896119894instead of 119899


infin

sum1198991 119899119903=0

Θ(120572)

1198991119899119903


119903

=infin

sum1198991 119899119903=0

119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2) (minus119910

119894)119896119894119905119899119894

119894


=infin

sum1198991 119899119903=0

119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2) (minus119910

119894)119896119894


11990511989911sdot sdot sdot 119905119899119903

119903

(42)





(119898119909)119899

119899=

[1198992]

sum119896=0

(] + 119899 minus 2119896) 119862]119899minus2119896

(1198981199092)

119896(])119899+1minus119896

(119898119909)119899

119899=

[1198992]

sum119896=0

119867119899minus2119896

(1198981199092)

119896 (119899 minus 2119896)

(119898119909)119899

119899= 2119899

119899

sum119896=0

(minus1)119896(120572 + 1)119899119871(120572)119896

(1198981199092)

(119899 minus 119896)(120572 + 1)119896

(43)


1198991 119899119903



119899(x y m) and

Some Properties



properties

51TheCase of119898119894= 2 119910

119894= 1 119894 = 1 2 119903 in (2) This case


(1 minus 211990911199051+ 1199052


119903119905119903+ 1199052

119903)minus120572

=infin

sum1198991 119899119903=0

119862(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(44)



119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0


)


119903 (119899

1minus 2119896

1)


119903))

minus1

)


119903)119899119903minus2119896119903

(45)


1198991 119899119903

(1199091 119909



119894(119894 = 1 2 119903)

Thus 119862(120572)

1198991 119899119903

(1199091 119909


1+


1 119909


also yields

119862(120572)

1198991 119899119903

(1199091 119909

119903)

=21198991+sdotsdotsdot+119899119903(120572)

1198991+sdotsdotsdot+119899119903


119903

11990911989911sdot sdot sdot 119909119899119903

119903+ Π (119909

1 119909

119903)

(46)

whereΠ(1199091 119909



119903minus



1199091rarr minus119909

1and 119905

1rarr minus119905

1 we have

119862(120572)

1198991 119899119903

(minus1199091 119909

2 119909

119903) = (minus1)1198991119862(120572)

1198991 119899119903

(1199091 119909

119903)

(47)


119862(120572)

1198991 119899119903

(1199091 119909


119894 119909

119894+1 119909

119903)

= (minus1)119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

(48)


119894and 119905


119894 119894 = 1 2 119903 in (44)

we obtain

119862(120572)

1198991 119899119903



1198991 119899119903

(1199091 119909

119903)

(49)


1198991 119899119903

(1199091 119909

119903) one has

119862(120572)

21198991 2119899119903


1198991+sdotsdotsdot+119899119903


119903

(50)


119862(120572)

1198991 119899119903

(0 0 0) = 0 (51)


(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991119899119903=0

119862(120572)

1198991 119899119903

(0 0 0) 11990511989911sdot sdot sdot 119905119899119903

119903

(52)


(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991 119899119903=0



119903(120572)

1198991+sdotsdotsdot+119899119903119905211989911

sdot sdot sdot 1199052119899119903119903

(53)



desired


1198991 119899119903

(1199091 119909

119903)


119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

= 119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus119903

sum119894=1

120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

(54)

hold for 119899119894ge 1 119894 = 1 2 119903


1198991 119899119903

(1199091 119909


119862(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119862(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119862(120573)

1198961 119896119903

(1199091 119909

119903)

(55)



1198991 119899119903

(1199091 119909



119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(2119899119894minus 4119896

119894minus 4119904

119894+ 1)

119896119894119904119894(32)

119899119894minus2119896119894minus119904119894


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(]119894+ 119899

119894minus 2119896

119894minus 2119904

119894)

119896119894119904119894(]

119894)119899119894minus2119896119894minus119904119894+1


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0


(119909119894)

119896119894119904119894 (119899

119894minus 2119896

119894minus 2119904

119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

119899119894minus2119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus2119896119894

2119899119894minus2119896119894(minus1)119904119894

119896119894 (119899

119894minus 2119896

119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus1)119896119894119871(120573119894)119904119894

(119909119894)


(56)


1198991 119899119903

(1199091 119909

119903) given by (44)


1198991 119899119903

(1199091


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

times 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

(max 1

11990921

1

1199092119903

lt 1)

(57)

where 119865(119903)119861


119865(119903)119861

[1198861 119886

119903 1198871 119887

119903 119888 119909

1 119909

119903]

=infin

sum1198991119899119903=0

(1198861)1198991

sdot sdot sdot (119886119903)119899119903

(1198871)1198991

sdot sdot sdot (119887119903)119899119903

(119888)1198991+sdotsdotsdot+119899119903

11990911989911


119909119899119903119903

119899119903

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091199031003816100381610038161003816 lt 1)

(58)


(120572)119899minus119896

=(minus1)119896(120572)

119899

(1 minus 120572 minus 119899)119896

(minus119899)2119896

= 22119896(minus119899

2)119896

(minus119899 + 1

2)119896

(119899 minus 2119896) =119899

(minus119899)2119896

(59)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0

(((120572)1198991+sdotsdotsdot+119899119903


2)1198961

times (minus119899

1+ 1

2)1198961


2)119896119903

times(minus119899

119903+ 1

2)119896119903

)


1199031198991 sdot sdot sdot 119899

119903


119903)1198961+sdotsdotsdot+119896119903

)minus1

)


119903)119899119903minus2119896119903

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903


times[11989912]

sum1198961=0


sum119896119903=0

(((minus1198991

2)1198961

(minus119899

1+ 1

2)1198961


2)119896119903

(minus119899

119903+ 1

2)119896119903

)


119903)1198961+sdotsdotsdot+119896119903


119903)

minus1

)

times(1

11990921

)1198961

sdot sdot sdot (1

1199092119903

)119896119903

= 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

times(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

(60)

wheremax 111990921 11199092



52 The Case of 119909119894= 0 119910

119894= minus119909

119894 119894 = 1 2 119903 in (2) This

case yields

(1 minus 119909111990511989811

minus 119909211990511989822


=infin

sum1198991 119899119903=0

119880(120572)

1198991119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(61)



1198991119899119903

(1199091 119909

119903) satisfy

119898119894119909119894

120597

120597119909119894

119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

= (119899119894minus 119898

119894+ 1)119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119898119894119909119894

120597

120597119909119894

times 119880(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times 119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

(62)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


mula

119880(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119880(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119880(120573)

1198961 119896119903

(1199091 119909

119903)

(63)




119903119905119903)minus120572

=infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(64)


119866(120572)

1198991 119899119903

(1199091 119909

119903) =

(120572)1198991+sdotsdotsdot+119899119903


11990311990911989911sdot sdot sdot 119909119899119903

119903 (65)


1198991 119899119903

(1199091 119909

119903) satisfy

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903) = 119909

119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903) (66)

for 119894 = 1 2 119903 and119903

sum119894=1

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903)

(67)


1198991 119899119903

(1199091 119909


119866(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119866(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119866(120573)

1198961 119896119903

(1199091 119909

119903)

(68)



1198991 119899119903

(1199091



119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(]119894+ 119899

119894minus 2119904

119894)

119904119894(]

119894)119899119894minus119904119894+1

119862]119894119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

119867119899119894minus2119904119894

(1199091198942)

119904119894 (119899

119894minus 2119904

119894) (120572)

1198991+sdotsdotsdot+119899119903

119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

119899119894

sum119904119894=0

(120573119894+ 1)

119899119894

2119899119894(minus1)119904119894

(119899119894minus 119904

119894)(120573

119894+ 1)

119904119894

119871(120573119894)119904119894

(119909119894

2)


(69)


1198991 119899119903

(1199091 119909

119903)


1198991 119899119903

(1199091 119909

119903) the


infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(70)

for each 119894 = 1 2 119903 where


0= 1 (71)

and g (120572120573)


defined by [14]


=infin

sum119899=0

g (120572120573)

119899(119909 119910) 119905119899


)

(72)


120585119894


119894 centered 120585

119894= 119911

119894and 120585


0 lt 120576119894lt 1003816100381610038161003816119909119894

1003816100381610038161003816minus1

10038161003816100381610038161003816100381610038161003816100381610038161 minus

119903

sum119896=1

119909119896119911119896

1003816100381610038161003816100381610038161003816100381610038161003816 (73)


119863119898

119911119894(1 minus 119909


119903119911119903)minus120572

119911119896

119894

=

119898

2120587119894

∮C120585119894

120585119896

119894(1 minus 119909



119903119911119903)minus120572

(120585119894minus 119911

119894)119898+1

119889120585119894

=

119898

2120587119894


(120585119894+ 119911

119894)119896


119894(120585119894+ 119911


119903119911119903)minus120572

120585119898+1

119894

119889120585119894

=

119898

2120587119894

119911119896

119894(1 minus 119909


119903119911119903)minus120572


(1 + 120585119894119911

119894)119896

120585119898+1

119894

(1 minus

119909119894120585119894


119903119911119903

)

minus120572

119889120585119894

= 119898119911119896

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894


119903119911119903

minus

1

119911119894

)

= 119898119911119896minus119898

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894119911119894


119903119911119903

minus1)

(74)


119863119898

119911119894


119903119911119903)minus120572

119911119896119894

= 119863119898

119911119894

119911119896119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903

=infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903)119889119898

119889119911119898119894

11991111989911sdot sdot sdot 119911119899119894+119896

119894sdot sdot sdot 119911119899119903

119903

= 119911119896minus119898119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903

(75)


infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572

times g(120572minus119896)119898

(119909119894119911119894


119903119911119903

minus1)

(76)


[119899119894+ 119898]

119898= (119899

119894+ 119898) (119899


119894+ 1)

= (119899119894+ 1)

119898

(77)



infin

sum1198991 119899119903=0

(119899119894+ 1)

119898119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(78)



(1 minus 11990911199051minus 119909


119903119905119903119903)minus120572

=infin

sum1198991119899119903=0

119867(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(79)



1198991 119899119903

(1199091 119909

119903) satisfy

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

= (119899119894minus 119894 + 1)119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

119903

sum119894=1

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119894 + 1)

times 119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

(80)

for 119899119894ge 119894 minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


119867(120572+120573)

1198991119899119903

(1199091 119909

119903)

=1198991

sum1198961=0


sum119896119903=0

119867(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119867(120573)

1198961 119896119903

(1199091 119909

119903)

(81)


= 119894 119909119894

= 0 119910119894

= minus119909119894


1198991 119899119903

(1199091 119909

119903)

Remark 28 If

Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)1199081198961

1sdot sdot sdot 119908119896119903

119903

(82)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 and

120582 = (1205821 120582

119903) 120578 = (120578

1 120578

119903)w = (119908

1 119908

119903) then we

have1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572)

1198991minus1198971 119899119903minus119897119903

(1199091 119909

119903)

times 119867(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

(83)

References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Θ(120572)

1198991 119899119903

(x ym)

=119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

119899119894minus119898119894119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus119898119894119896119894

2119899119894minus119898119894119896119894(minus1)119904119894

119896119894 (119899

119894minus 119898

119894119896119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus119910119894)119896119894119871(120573119894)

119904119894

(119898119894119909119894

2)


(38)

Proof By (2) we get

infin

sum1198991119899119903=0

Θ(120572)

1198991 119899119903


119903

= (1 minus119903

sum119894=1

(119898119894119909119894119905119894minus 119910

119894119905119898119894119894))

minus120572

=infin

sum1198991 119899119903=0

infin

sum1198961 119896119903=0



1199031198991 sdot sdot sdot 119899



119903119909119903)1198991199031199101198961

1sdot sdot sdot 119910119896119903

119903

times 1199051198991+119898111989611

sdot sdot sdot 119905119899119903+119898119903119896119903119903

(39)


(119898119909)119899

119899=

[1198992]

sum119904=0

(2119899 minus 4119904 + 1) 119875119899minus2119904

(1198981199092)

119904(32)119899minus119904

(40)

we can write that

infin

sum1198991119899119903=0

Θ(120572)

1198991 119899119903


119903

=119903

prod119894=1

infin

sum119899119894=0

infin

sum119896119894=0

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119898119894119909119894

2)

times(minus119910119894)119896119894119905119899119894+119898119894119896119894

119894


(41)

Getting 119899119894minus 119898

119894119896119894instead of 119899


infin

sum1198991 119899119903=0

Θ(120572)

1198991119899119903


119903

=infin

sum1198991 119899119903=0

119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2) (minus119910

119894)119896119894119905119899119894

119894


=infin

sum1198991 119899119903=0

119903

prod119894=1

[119899119894119898119894]

sum119896119894=0

[(119899119894minus119898119894119896119894)2]

sum119904119894=0

(2119899119894minus 2119898

119894119896119894minus 4119904

119894+ 1)

119904119894119896119894(32)

119899119894minus119898119894119896119894minus119904119894


(119898119894119909119894

2) (minus119910

119894)119896119894


11990511989911sdot sdot sdot 119905119899119903

119903

(42)





(119898119909)119899

119899=

[1198992]

sum119896=0

(] + 119899 minus 2119896) 119862]119899minus2119896

(1198981199092)

119896(])119899+1minus119896

(119898119909)119899

119899=

[1198992]

sum119896=0

119867119899minus2119896

(1198981199092)

119896 (119899 minus 2119896)

(119898119909)119899

119899= 2119899

119899

sum119896=0

(minus1)119896(120572 + 1)119899119871(120572)119896

(1198981199092)

(119899 minus 119896)(120572 + 1)119896

(43)


1198991 119899119903



119899(x y m) and

Some Properties



properties

51TheCase of119898119894= 2 119910

119894= 1 119894 = 1 2 119903 in (2) This case


(1 minus 211990911199051+ 1199052


119903119905119903+ 1199052

119903)minus120572

=infin

sum1198991 119899119903=0

119862(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(44)



119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0


)


119903 (119899

1minus 2119896

1)


119903))

minus1

)


119903)119899119903minus2119896119903

(45)


1198991 119899119903

(1199091 119909



119894(119894 = 1 2 119903)

Thus 119862(120572)

1198991 119899119903

(1199091 119909


1+


1 119909


also yields

119862(120572)

1198991 119899119903

(1199091 119909

119903)

=21198991+sdotsdotsdot+119899119903(120572)

1198991+sdotsdotsdot+119899119903


119903

11990911989911sdot sdot sdot 119909119899119903

119903+ Π (119909

1 119909

119903)

(46)

whereΠ(1199091 119909



119903minus



1199091rarr minus119909

1and 119905

1rarr minus119905

1 we have

119862(120572)

1198991 119899119903

(minus1199091 119909

2 119909

119903) = (minus1)1198991119862(120572)

1198991 119899119903

(1199091 119909

119903)

(47)


119862(120572)

1198991 119899119903

(1199091 119909


119894 119909

119894+1 119909

119903)

= (minus1)119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

(48)


119894and 119905


119894 119894 = 1 2 119903 in (44)

we obtain

119862(120572)

1198991 119899119903



1198991 119899119903

(1199091 119909

119903)

(49)


1198991 119899119903

(1199091 119909

119903) one has

119862(120572)

21198991 2119899119903


1198991+sdotsdotsdot+119899119903


119903

(50)


119862(120572)

1198991 119899119903

(0 0 0) = 0 (51)


(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991119899119903=0

119862(120572)

1198991 119899119903

(0 0 0) 11990511989911sdot sdot sdot 119905119899119903

119903

(52)


(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991 119899119903=0



119903(120572)

1198991+sdotsdotsdot+119899119903119905211989911

sdot sdot sdot 1199052119899119903119903

(53)



desired


1198991 119899119903

(1199091 119909

119903)


119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

= 119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus119903

sum119894=1

120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

(54)

hold for 119899119894ge 1 119894 = 1 2 119903


1198991 119899119903

(1199091 119909


119862(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119862(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119862(120573)

1198961 119896119903

(1199091 119909

119903)

(55)



1198991 119899119903

(1199091 119909



119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(2119899119894minus 4119896

119894minus 4119904

119894+ 1)

119896119894119904119894(32)

119899119894minus2119896119894minus119904119894


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(]119894+ 119899

119894minus 2119896

119894minus 2119904

119894)

119896119894119904119894(]

119894)119899119894minus2119896119894minus119904119894+1


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0


(119909119894)

119896119894119904119894 (119899

119894minus 2119896

119894minus 2119904

119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

119899119894minus2119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus2119896119894

2119899119894minus2119896119894(minus1)119904119894

119896119894 (119899

119894minus 2119896

119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus1)119896119894119871(120573119894)119904119894

(119909119894)


(56)


1198991 119899119903

(1199091 119909

119903) given by (44)


1198991 119899119903

(1199091


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

times 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

(max 1

11990921

1

1199092119903

lt 1)

(57)

where 119865(119903)119861


119865(119903)119861

[1198861 119886

119903 1198871 119887

119903 119888 119909

1 119909

119903]

=infin

sum1198991119899119903=0

(1198861)1198991

sdot sdot sdot (119886119903)119899119903

(1198871)1198991

sdot sdot sdot (119887119903)119899119903

(119888)1198991+sdotsdotsdot+119899119903

11990911989911


119909119899119903119903

119899119903

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091199031003816100381610038161003816 lt 1)

(58)


(120572)119899minus119896

=(minus1)119896(120572)

119899

(1 minus 120572 minus 119899)119896

(minus119899)2119896

= 22119896(minus119899

2)119896

(minus119899 + 1

2)119896

(119899 minus 2119896) =119899

(minus119899)2119896

(59)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0

(((120572)1198991+sdotsdotsdot+119899119903


2)1198961

times (minus119899

1+ 1

2)1198961


2)119896119903

times(minus119899

119903+ 1

2)119896119903

)


1199031198991 sdot sdot sdot 119899

119903


119903)1198961+sdotsdotsdot+119896119903

)minus1

)


119903)119899119903minus2119896119903

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903


times[11989912]

sum1198961=0


sum119896119903=0

(((minus1198991

2)1198961

(minus119899

1+ 1

2)1198961


2)119896119903

(minus119899

119903+ 1

2)119896119903

)


119903)1198961+sdotsdotsdot+119896119903


119903)

minus1

)

times(1

11990921

)1198961

sdot sdot sdot (1

1199092119903

)119896119903

= 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

times(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

(60)

wheremax 111990921 11199092



52 The Case of 119909119894= 0 119910

119894= minus119909

119894 119894 = 1 2 119903 in (2) This

case yields

(1 minus 119909111990511989811

minus 119909211990511989822


=infin

sum1198991 119899119903=0

119880(120572)

1198991119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(61)



1198991119899119903

(1199091 119909

119903) satisfy

119898119894119909119894

120597

120597119909119894

119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

= (119899119894minus 119898

119894+ 1)119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119898119894119909119894

120597

120597119909119894

times 119880(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times 119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

(62)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


mula

119880(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119880(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119880(120573)

1198961 119896119903

(1199091 119909

119903)

(63)




119903119905119903)minus120572

=infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(64)


119866(120572)

1198991 119899119903

(1199091 119909

119903) =

(120572)1198991+sdotsdotsdot+119899119903


11990311990911989911sdot sdot sdot 119909119899119903

119903 (65)


1198991 119899119903

(1199091 119909

119903) satisfy

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903) = 119909

119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903) (66)

for 119894 = 1 2 119903 and119903

sum119894=1

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903)

(67)


1198991 119899119903

(1199091 119909


119866(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119866(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119866(120573)

1198961 119896119903

(1199091 119909

119903)

(68)



1198991 119899119903

(1199091



119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(]119894+ 119899

119894minus 2119904

119894)

119904119894(]

119894)119899119894minus119904119894+1

119862]119894119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

119867119899119894minus2119904119894

(1199091198942)

119904119894 (119899

119894minus 2119904

119894) (120572)

1198991+sdotsdotsdot+119899119903

119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

119899119894

sum119904119894=0

(120573119894+ 1)

119899119894

2119899119894(minus1)119904119894

(119899119894minus 119904

119894)(120573

119894+ 1)

119904119894

119871(120573119894)119904119894

(119909119894

2)


(69)


1198991 119899119903

(1199091 119909

119903)


1198991 119899119903

(1199091 119909

119903) the


infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(70)

for each 119894 = 1 2 119903 where


0= 1 (71)

and g (120572120573)


defined by [14]


=infin

sum119899=0

g (120572120573)

119899(119909 119910) 119905119899


)

(72)


120585119894


119894 centered 120585

119894= 119911

119894and 120585


0 lt 120576119894lt 1003816100381610038161003816119909119894

1003816100381610038161003816minus1

10038161003816100381610038161003816100381610038161003816100381610038161 minus

119903

sum119896=1

119909119896119911119896

1003816100381610038161003816100381610038161003816100381610038161003816 (73)


119863119898

119911119894(1 minus 119909


119903119911119903)minus120572

119911119896

119894

=

119898

2120587119894

∮C120585119894

120585119896

119894(1 minus 119909



119903119911119903)minus120572

(120585119894minus 119911

119894)119898+1

119889120585119894

=

119898

2120587119894


(120585119894+ 119911

119894)119896


119894(120585119894+ 119911


119903119911119903)minus120572

120585119898+1

119894

119889120585119894

=

119898

2120587119894

119911119896

119894(1 minus 119909


119903119911119903)minus120572


(1 + 120585119894119911

119894)119896

120585119898+1

119894

(1 minus

119909119894120585119894


119903119911119903

)

minus120572

119889120585119894

= 119898119911119896

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894


119903119911119903

minus

1

119911119894

)

= 119898119911119896minus119898

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894119911119894


119903119911119903

minus1)

(74)


119863119898

119911119894


119903119911119903)minus120572

119911119896119894

= 119863119898

119911119894

119911119896119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903

=infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903)119889119898

119889119911119898119894

11991111989911sdot sdot sdot 119911119899119894+119896

119894sdot sdot sdot 119911119899119903

119903

= 119911119896minus119898119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903

(75)


infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572

times g(120572minus119896)119898

(119909119894119911119894


119903119911119903

minus1)

(76)


[119899119894+ 119898]

119898= (119899

119894+ 119898) (119899


119894+ 1)

= (119899119894+ 1)

119898

(77)



infin

sum1198991 119899119903=0

(119899119894+ 1)

119898119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(78)



(1 minus 11990911199051minus 119909


119903119905119903119903)minus120572

=infin

sum1198991119899119903=0

119867(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(79)



1198991 119899119903

(1199091 119909

119903) satisfy

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

= (119899119894minus 119894 + 1)119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

119903

sum119894=1

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119894 + 1)

times 119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

(80)

for 119899119894ge 119894 minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


119867(120572+120573)

1198991119899119903

(1199091 119909

119903)

=1198991

sum1198961=0


sum119896119903=0

119867(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119867(120573)

1198961 119896119903

(1199091 119909

119903)

(81)


= 119894 119909119894

= 0 119910119894

= minus119909119894


1198991 119899119903

(1199091 119909

119903)

Remark 28 If

Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)1199081198961

1sdot sdot sdot 119908119896119903

119903

(82)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 and

120582 = (1205821 120582

119903) 120578 = (120578

1 120578

119903)w = (119908

1 119908

119903) then we

have1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572)

1198991minus1198971 119899119903minus119897119903

(1199091 119909

119903)

times 119867(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

(83)

References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0


)


119903 (119899

1minus 2119896

1)


119903))

minus1

)


119903)119899119903minus2119896119903

(45)


1198991 119899119903

(1199091 119909



119894(119894 = 1 2 119903)

Thus 119862(120572)

1198991 119899119903

(1199091 119909


1+


1 119909


also yields

119862(120572)

1198991 119899119903

(1199091 119909

119903)

=21198991+sdotsdotsdot+119899119903(120572)

1198991+sdotsdotsdot+119899119903


119903

11990911989911sdot sdot sdot 119909119899119903

119903+ Π (119909

1 119909

119903)

(46)

whereΠ(1199091 119909



119903minus



1199091rarr minus119909

1and 119905

1rarr minus119905

1 we have

119862(120572)

1198991 119899119903

(minus1199091 119909

2 119909

119903) = (minus1)1198991119862(120572)

1198991 119899119903

(1199091 119909

119903)

(47)


119862(120572)

1198991 119899119903

(1199091 119909


119894 119909

119894+1 119909

119903)

= (minus1)119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

(48)


119894and 119905


119894 119894 = 1 2 119903 in (44)

we obtain

119862(120572)

1198991 119899119903



1198991 119899119903

(1199091 119909

119903)

(49)


1198991 119899119903

(1199091 119909

119903) one has

119862(120572)

21198991 2119899119903


1198991+sdotsdotsdot+119899119903


119903

(50)


119862(120572)

1198991 119899119903

(0 0 0) = 0 (51)


(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991119899119903=0

119862(120572)

1198991 119899119903

(0 0 0) 11990511989911sdot sdot sdot 119905119899119903

119903

(52)


(1 + 11990521+ sdot sdot sdot + 1199052

119903)minus120572

=infin

sum1198991 119899119903=0



119903(120572)

1198991+sdotsdotsdot+119899119903119905211989911

sdot sdot sdot 1199052119899119903119903

(53)



desired


1198991 119899119903

(1199091 119909

119903)


119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

= 119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119899119894119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119862(120572)

1198991 119899119903

(1199091 119909

119903)

minus119903

sum119894=1

120597

120597119909119894

119862(120572)

1198991 119899119894minus1119899119894minus1119899119894+1119899119903

(1199091 119909

119903)

(54)

hold for 119899119894ge 1 119894 = 1 2 119903


1198991 119899119903

(1199091 119909


119862(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119862(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119862(120573)

1198961 119896119903

(1199091 119909

119903)

(55)



1198991 119899119903

(1199091 119909



119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(2119899119894minus 4119896

119894minus 4119904

119894+ 1)

119896119894119904119894(32)

119899119894minus2119896119894minus119904119894


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(]119894+ 119899

119894minus 2119896

119894minus 2119904

119894)

119896119894119904119894(]

119894)119899119894minus2119896119894minus119904119894+1


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0


(119909119894)

119896119894119904119894 (119899

119894minus 2119896

119894minus 2119904

119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

119899119894minus2119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus2119896119894

2119899119894minus2119896119894(minus1)119904119894

119896119894 (119899

119894minus 2119896

119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus1)119896119894119871(120573119894)119904119894

(119909119894)


(56)


1198991 119899119903

(1199091 119909

119903) given by (44)


1198991 119899119903

(1199091


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

times 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

(max 1

11990921

1

1199092119903

lt 1)

(57)

where 119865(119903)119861


119865(119903)119861

[1198861 119886

119903 1198871 119887

119903 119888 119909

1 119909

119903]

=infin

sum1198991119899119903=0

(1198861)1198991

sdot sdot sdot (119886119903)119899119903

(1198871)1198991

sdot sdot sdot (119887119903)119899119903

(119888)1198991+sdotsdotsdot+119899119903

11990911989911


119909119899119903119903

119899119903

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091199031003816100381610038161003816 lt 1)

(58)


(120572)119899minus119896

=(minus1)119896(120572)

119899

(1 minus 120572 minus 119899)119896

(minus119899)2119896

= 22119896(minus119899

2)119896

(minus119899 + 1

2)119896

(119899 minus 2119896) =119899

(minus119899)2119896

(59)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0

(((120572)1198991+sdotsdotsdot+119899119903


2)1198961

times (minus119899

1+ 1

2)1198961


2)119896119903

times(minus119899

119903+ 1

2)119896119903

)


1199031198991 sdot sdot sdot 119899

119903


119903)1198961+sdotsdotsdot+119896119903

)minus1

)


119903)119899119903minus2119896119903

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903


times[11989912]

sum1198961=0


sum119896119903=0

(((minus1198991

2)1198961

(minus119899

1+ 1

2)1198961


2)119896119903

(minus119899

119903+ 1

2)119896119903

)


119903)1198961+sdotsdotsdot+119896119903


119903)

minus1

)

times(1

11990921

)1198961

sdot sdot sdot (1

1199092119903

)119896119903

= 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

times(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

(60)

wheremax 111990921 11199092



52 The Case of 119909119894= 0 119910

119894= minus119909

119894 119894 = 1 2 119903 in (2) This

case yields

(1 minus 119909111990511989811

minus 119909211990511989822


=infin

sum1198991 119899119903=0

119880(120572)

1198991119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(61)



1198991119899119903

(1199091 119909

119903) satisfy

119898119894119909119894

120597

120597119909119894

119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

= (119899119894minus 119898

119894+ 1)119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119898119894119909119894

120597

120597119909119894

times 119880(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times 119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

(62)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


mula

119880(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119880(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119880(120573)

1198961 119896119903

(1199091 119909

119903)

(63)




119903119905119903)minus120572

=infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(64)


119866(120572)

1198991 119899119903

(1199091 119909

119903) =

(120572)1198991+sdotsdotsdot+119899119903


11990311990911989911sdot sdot sdot 119909119899119903

119903 (65)


1198991 119899119903

(1199091 119909

119903) satisfy

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903) = 119909

119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903) (66)

for 119894 = 1 2 119903 and119903

sum119894=1

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903)

(67)


1198991 119899119903

(1199091 119909


119866(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119866(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119866(120573)

1198961 119896119903

(1199091 119909

119903)

(68)



1198991 119899119903

(1199091



119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(]119894+ 119899

119894minus 2119904

119894)

119904119894(]

119894)119899119894minus119904119894+1

119862]119894119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

119867119899119894minus2119904119894

(1199091198942)

119904119894 (119899

119894minus 2119904

119894) (120572)

1198991+sdotsdotsdot+119899119903

119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

119899119894

sum119904119894=0

(120573119894+ 1)

119899119894

2119899119894(minus1)119904119894

(119899119894minus 119904

119894)(120573

119894+ 1)

119904119894

119871(120573119894)119904119894

(119909119894

2)


(69)


1198991 119899119903

(1199091 119909

119903)


1198991 119899119903

(1199091 119909

119903) the


infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(70)

for each 119894 = 1 2 119903 where


0= 1 (71)

and g (120572120573)


defined by [14]


=infin

sum119899=0

g (120572120573)

119899(119909 119910) 119905119899


)

(72)


120585119894


119894 centered 120585

119894= 119911

119894and 120585


0 lt 120576119894lt 1003816100381610038161003816119909119894

1003816100381610038161003816minus1

10038161003816100381610038161003816100381610038161003816100381610038161 minus

119903

sum119896=1

119909119896119911119896

1003816100381610038161003816100381610038161003816100381610038161003816 (73)


119863119898

119911119894(1 minus 119909


119903119911119903)minus120572

119911119896

119894

=

119898

2120587119894

∮C120585119894

120585119896

119894(1 minus 119909



119903119911119903)minus120572

(120585119894minus 119911

119894)119898+1

119889120585119894

=

119898

2120587119894


(120585119894+ 119911

119894)119896


119894(120585119894+ 119911


119903119911119903)minus120572

120585119898+1

119894

119889120585119894

=

119898

2120587119894

119911119896

119894(1 minus 119909


119903119911119903)minus120572


(1 + 120585119894119911

119894)119896

120585119898+1

119894

(1 minus

119909119894120585119894


119903119911119903

)

minus120572

119889120585119894

= 119898119911119896

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894


119903119911119903

minus

1

119911119894

)

= 119898119911119896minus119898

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894119911119894


119903119911119903

minus1)

(74)


119863119898

119911119894


119903119911119903)minus120572

119911119896119894

= 119863119898

119911119894

119911119896119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903

=infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903)119889119898

119889119911119898119894

11991111989911sdot sdot sdot 119911119899119894+119896

119894sdot sdot sdot 119911119899119903

119903

= 119911119896minus119898119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903

(75)


infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572

times g(120572minus119896)119898

(119909119894119911119894


119903119911119903

minus1)

(76)


[119899119894+ 119898]

119898= (119899

119894+ 119898) (119899


119894+ 1)

= (119899119894+ 1)

119898

(77)



infin

sum1198991 119899119903=0

(119899119894+ 1)

119898119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(78)



(1 minus 11990911199051minus 119909


119903119905119903119903)minus120572

=infin

sum1198991119899119903=0

119867(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(79)



1198991 119899119903

(1199091 119909

119903) satisfy

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

= (119899119894minus 119894 + 1)119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

119903

sum119894=1

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119894 + 1)

times 119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

(80)

for 119899119894ge 119894 minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


119867(120572+120573)

1198991119899119903

(1199091 119909

119903)

=1198991

sum1198961=0


sum119896119903=0

119867(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119867(120573)

1198961 119896119903

(1199091 119909

119903)

(81)


= 119894 119909119894

= 0 119910119894

= minus119909119894


1198991 119899119903

(1199091 119909

119903)

Remark 28 If

Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)1199081198961

1sdot sdot sdot 119908119896119903

119903

(82)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 and

120582 = (1205821 120582

119903) 120578 = (120578

1 120578

119903)w = (119908

1 119908

119903) then we

have1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572)

1198991minus1198971 119899119903minus119897119903

(1199091 119909

119903)

times 119867(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

(83)

References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198991 119899119903

(1199091 119909



119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(2119899119894minus 4119896

119894minus 4119904

119894+ 1)

119896119894119904119894(32)

119899119894minus2119896119894minus119904119894


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0

(]119894+ 119899

119894minus 2119896

119894minus 2119904

119894)

119896119894119904119894(]

119894)119899119894minus2119896119894minus119904119894+1


(119909119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

[(119899119894minus2119896119894)2]

sum119904119894=0


(119909119894)

119896119894119904119894 (119899

119894minus 2119896

119894minus 2119904

119894)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119896119894=0

119899119894minus2119896119894

sum119904119894=0

(120573119894+ 1)

119899119894minus2119896119894

2119899119894minus2119896119894(minus1)119904119894

119896119894 (119899

119894minus 2119896

119894minus 119904

119894)(120573

119894+ 1)

119904119894

times(minus1)119896119894119871(120573119894)119904119894

(119909119894)


(56)


1198991 119899119903

(1199091 119909

119903) given by (44)


1198991 119899119903

(1199091


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

times 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

(max 1

11990921

1

1199092119903

lt 1)

(57)

where 119865(119903)119861


119865(119903)119861

[1198861 119886

119903 1198871 119887

119903 119888 119909

1 119909

119903]

=infin

sum1198991119899119903=0

(1198861)1198991

sdot sdot sdot (119886119903)119899119903

(1198871)1198991

sdot sdot sdot (119887119903)119899119903

(119888)1198991+sdotsdotsdot+119899119903

11990911989911


119909119899119903119903

119899119903

(max 100381610038161003816100381611990911003816100381610038161003816

10038161003816100381610038161199091199031003816100381610038161003816 lt 1)

(58)


(120572)119899minus119896

=(minus1)119896(120572)

119899

(1 minus 120572 minus 119899)119896

(minus119899)2119896

= 22119896(minus119899

2)119896

(minus119899 + 1

2)119896

(119899 minus 2119896) =119899

(minus119899)2119896

(59)


119862(120572)

1198991 119899119903

(1199091 119909

119903)

=[11989912]

sum1198961=0


sum119896119903=0

(((120572)1198991+sdotsdotsdot+119899119903


2)1198961

times (minus119899

1+ 1

2)1198961


2)119896119903

times(minus119899

119903+ 1

2)119896119903

)


1199031198991 sdot sdot sdot 119899

119903


119903)1198961+sdotsdotsdot+119896119903

)minus1

)


119903)119899119903minus2119896119903

=(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903


times[11989912]

sum1198961=0


sum119896119903=0

(((minus1198991

2)1198961

(minus119899

1+ 1

2)1198961


2)119896119903

(minus119899

119903+ 1

2)119896119903

)


119903)1198961+sdotsdotsdot+119896119903


119903)

minus1

)

times(1

11990921

)1198961

sdot sdot sdot (1

1199092119903

)119896119903

= 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

times(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

(60)

wheremax 111990921 11199092



52 The Case of 119909119894= 0 119910

119894= minus119909

119894 119894 = 1 2 119903 in (2) This

case yields

(1 minus 119909111990511989811

minus 119909211990511989822


=infin

sum1198991 119899119903=0

119880(120572)

1198991119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(61)



1198991119899119903

(1199091 119909

119903) satisfy

119898119894119909119894

120597

120597119909119894

119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

= (119899119894minus 119898

119894+ 1)119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119898119894119909119894

120597

120597119909119894

times 119880(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times 119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

(62)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


mula

119880(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119880(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119880(120573)

1198961 119896119903

(1199091 119909

119903)

(63)




119903119905119903)minus120572

=infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(64)


119866(120572)

1198991 119899119903

(1199091 119909

119903) =

(120572)1198991+sdotsdotsdot+119899119903


11990311990911989911sdot sdot sdot 119909119899119903

119903 (65)


1198991 119899119903

(1199091 119909

119903) satisfy

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903) = 119909

119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903) (66)

for 119894 = 1 2 119903 and119903

sum119894=1

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903)

(67)


1198991 119899119903

(1199091 119909


119866(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119866(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119866(120573)

1198961 119896119903

(1199091 119909

119903)

(68)



1198991 119899119903

(1199091



119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(]119894+ 119899

119894minus 2119904

119894)

119904119894(]

119894)119899119894minus119904119894+1

119862]119894119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

119867119899119894minus2119904119894

(1199091198942)

119904119894 (119899

119894minus 2119904

119894) (120572)

1198991+sdotsdotsdot+119899119903

119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

119899119894

sum119904119894=0

(120573119894+ 1)

119899119894

2119899119894(minus1)119904119894

(119899119894minus 119904

119894)(120573

119894+ 1)

119904119894

119871(120573119894)119904119894

(119909119894

2)


(69)


1198991 119899119903

(1199091 119909

119903)


1198991 119899119903

(1199091 119909

119903) the


infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(70)

for each 119894 = 1 2 119903 where


0= 1 (71)

and g (120572120573)


defined by [14]


=infin

sum119899=0

g (120572120573)

119899(119909 119910) 119905119899


)

(72)


120585119894


119894 centered 120585

119894= 119911

119894and 120585


0 lt 120576119894lt 1003816100381610038161003816119909119894

1003816100381610038161003816minus1

10038161003816100381610038161003816100381610038161003816100381610038161 minus

119903

sum119896=1

119909119896119911119896

1003816100381610038161003816100381610038161003816100381610038161003816 (73)


119863119898

119911119894(1 minus 119909


119903119911119903)minus120572

119911119896

119894

=

119898

2120587119894

∮C120585119894

120585119896

119894(1 minus 119909



119903119911119903)minus120572

(120585119894minus 119911

119894)119898+1

119889120585119894

=

119898

2120587119894


(120585119894+ 119911

119894)119896


119894(120585119894+ 119911


119903119911119903)minus120572

120585119898+1

119894

119889120585119894

=

119898

2120587119894

119911119896

119894(1 minus 119909


119903119911119903)minus120572


(1 + 120585119894119911

119894)119896

120585119898+1

119894

(1 minus

119909119894120585119894


119903119911119903

)

minus120572

119889120585119894

= 119898119911119896

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894


119903119911119903

minus

1

119911119894

)

= 119898119911119896minus119898

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894119911119894


119903119911119903

minus1)

(74)


119863119898

119911119894


119903119911119903)minus120572

119911119896119894

= 119863119898

119911119894

119911119896119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903

=infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903)119889119898

119889119911119898119894

11991111989911sdot sdot sdot 119911119899119894+119896

119894sdot sdot sdot 119911119899119903

119903

= 119911119896minus119898119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903

(75)


infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572

times g(120572minus119896)119898

(119909119894119911119894


119903119911119903

minus1)

(76)


[119899119894+ 119898]

119898= (119899

119894+ 119898) (119899


119894+ 1)

= (119899119894+ 1)

119898

(77)



infin

sum1198991 119899119903=0

(119899119894+ 1)

119898119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(78)



(1 minus 11990911199051minus 119909


119903119905119903119903)minus120572

=infin

sum1198991119899119903=0

119867(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(79)



1198991 119899119903

(1199091 119909

119903) satisfy

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

= (119899119894minus 119894 + 1)119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

119903

sum119894=1

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119894 + 1)

times 119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

(80)

for 119899119894ge 119894 minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


119867(120572+120573)

1198991119899119903

(1199091 119909

119903)

=1198991

sum1198961=0


sum119896119903=0

119867(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119867(120573)

1198961 119896119903

(1199091 119909

119903)

(81)


= 119894 119909119894

= 0 119910119894

= minus119909119894


1198991 119899119903

(1199091 119909

119903)

Remark 28 If

Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)1199081198961

1sdot sdot sdot 119908119896119903

119903

(82)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 and

120582 = (1205821 120582

119903) 120578 = (120578

1 120578

119903)w = (119908

1 119908

119903) then we

have1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572)

1198991minus1198971 119899119903minus119897119903

(1199091 119909

119903)

times 119867(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

(83)

References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times[11989912]

sum1198961=0


sum119896119903=0

(((minus1198991

2)1198961

(minus119899

1+ 1

2)1198961


2)119896119903

(minus119899

119903+ 1

2)119896119903

)


119903)1198961+sdotsdotsdot+119896119903


119903)

minus1

)

times(1

11990921

)1198961

sdot sdot sdot (1

1199092119903

)119896119903

= 119865(119903)119861

[ minus1198991

2 minus

119899119903

2minus119899

1+ 1

2

minus119899119903+ 1

2


1199031

11990921

1

1199092119903

]

times(120572)

1198991+sdotsdotsdot+119899119903


119903(2119909

1)1198991 sdot sdot sdot (2119909

119903)119899119903

(60)

wheremax 111990921 11199092



52 The Case of 119909119894= 0 119910

119894= minus119909

119894 119894 = 1 2 119903 in (2) This

case yields

(1 minus 119909111990511989811

minus 119909211990511989822


=infin

sum1198991 119899119903=0

119880(120572)

1198991119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(61)



1198991119899119903

(1199091 119909

119903) satisfy

119898119894119909119894

120597

120597119909119894

119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

= (119899119894minus 119898

119894+ 1)119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

119903

sum119894=1

119898119894119909119894

120597

120597119909119894

times 119880(120572)

1198991 119899119894minus1119899119894minus119898119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119898

119894+ 1)

times 119880(120572)

1198991119899119894minus1119899119894minus119898119894+1119899119894+1119899119903

(1199091 119909

119903)

(62)

for 119899119894ge 119898

119894minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


mula

119880(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119880(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119880(120573)

1198961 119896119903

(1199091 119909

119903)

(63)




119903119905119903)minus120572

=infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(64)


119866(120572)

1198991 119899119903

(1199091 119909

119903) =

(120572)1198991+sdotsdotsdot+119899119903


11990311990911989911sdot sdot sdot 119909119899119903

119903 (65)


1198991 119899119903

(1199091 119909

119903) satisfy

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903) = 119909

119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903) (66)

for 119894 = 1 2 119903 and119903

sum119894=1

119899119894119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

sum119894=1

119909119894

120597

120597119909119894

119866(120572)

1198991 119899119903

(1199091 119909

119903)

(67)


1198991 119899119903

(1199091 119909


119866(120572+120573)

1198991 119899119903

(1199091 119909

119903) =

1198991

sum1198961=0


sum119896119903=0

119866(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119866(120573)

1198961 119896119903

(1199091 119909

119903)

(68)



1198991 119899119903

(1199091



119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(]119894+ 119899

119894minus 2119904

119894)

119904119894(]

119894)119899119894minus119904119894+1

119862]119894119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

119867119899119894minus2119904119894

(1199091198942)

119904119894 (119899

119894minus 2119904

119894) (120572)

1198991+sdotsdotsdot+119899119903

119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

119899119894

sum119904119894=0

(120573119894+ 1)

119899119894

2119899119894(minus1)119904119894

(119899119894minus 119904

119894)(120573

119894+ 1)

119904119894

119871(120573119894)119904119894

(119909119894

2)


(69)


1198991 119899119903

(1199091 119909

119903)


1198991 119899119903

(1199091 119909

119903) the


infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(70)

for each 119894 = 1 2 119903 where


0= 1 (71)

and g (120572120573)


defined by [14]


=infin

sum119899=0

g (120572120573)

119899(119909 119910) 119905119899


)

(72)


120585119894


119894 centered 120585

119894= 119911

119894and 120585


0 lt 120576119894lt 1003816100381610038161003816119909119894

1003816100381610038161003816minus1

10038161003816100381610038161003816100381610038161003816100381610038161 minus

119903

sum119896=1

119909119896119911119896

1003816100381610038161003816100381610038161003816100381610038161003816 (73)


119863119898

119911119894(1 minus 119909


119903119911119903)minus120572

119911119896

119894

=

119898

2120587119894

∮C120585119894

120585119896

119894(1 minus 119909



119903119911119903)minus120572

(120585119894minus 119911

119894)119898+1

119889120585119894

=

119898

2120587119894


(120585119894+ 119911

119894)119896


119894(120585119894+ 119911


119903119911119903)minus120572

120585119898+1

119894

119889120585119894

=

119898

2120587119894

119911119896

119894(1 minus 119909


119903119911119903)minus120572


(1 + 120585119894119911

119894)119896

120585119898+1

119894

(1 minus

119909119894120585119894


119903119911119903

)

minus120572

119889120585119894

= 119898119911119896

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894


119903119911119903

minus

1

119911119894

)

= 119898119911119896minus119898

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894119911119894


119903119911119903

minus1)

(74)


119863119898

119911119894


119903119911119903)minus120572

119911119896119894

= 119863119898

119911119894

119911119896119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903

=infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903)119889119898

119889119911119898119894

11991111989911sdot sdot sdot 119911119899119894+119896

119894sdot sdot sdot 119911119899119903

119903

= 119911119896minus119898119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903

(75)


infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572

times g(120572minus119896)119898

(119909119894119911119894


119903119911119903

minus1)

(76)


[119899119894+ 119898]

119898= (119899

119894+ 119898) (119899


119894+ 1)

= (119899119894+ 1)

119898

(77)



infin

sum1198991 119899119903=0

(119899119894+ 1)

119898119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(78)



(1 minus 11990911199051minus 119909


119903119905119903119903)minus120572

=infin

sum1198991119899119903=0

119867(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(79)



1198991 119899119903

(1199091 119909

119903) satisfy

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

= (119899119894minus 119894 + 1)119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

119903

sum119894=1

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119894 + 1)

times 119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

(80)

for 119899119894ge 119894 minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


119867(120572+120573)

1198991119899119903

(1199091 119909

119903)

=1198991

sum1198961=0


sum119896119903=0

119867(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119867(120573)

1198961 119896119903

(1199091 119909

119903)

(81)


= 119894 119909119894

= 0 119910119894

= minus119909119894


1198991 119899119903

(1199091 119909

119903)

Remark 28 If

Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)1199081198961

1sdot sdot sdot 119908119896119903

119903

(82)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 and

120582 = (1205821 120582

119903) 120578 = (120578

1 120578

119903)w = (119908

1 119908

119903) then we

have1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572)

1198991minus1198971 119899119903minus119897119903

(1199091 119909

119903)

times 119867(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

(83)

References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198991 119899119903

(1199091



119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(2119899119894minus 4119904

119894+ 1)

119904119894(32)

119899119894minus119904119894

119875119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

(]119894+ 119899

119894minus 2119904

119894)

119904119894(]

119894)119899119894minus119904119894+1

119862]119894119899119894minus2119904119894

(119909119894

2)


119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

[1198991198942]

sum119904119894=0

119867119899119894minus2119904119894

(1199091198942)

119904119894 (119899

119894minus 2119904

119894) (120572)

1198991+sdotsdotsdot+119899119903

119866(120572)

1198991 119899119903

(1199091 119909

119903)

=119903

prod119894=1

119899119894

sum119904119894=0

(120573119894+ 1)

119899119894

2119899119894(minus1)119904119894

(119899119894minus 119904

119894)(120573

119894+ 1)

119904119894

119871(120573119894)119904119894

(119909119894

2)


(69)


1198991 119899119903

(1199091 119909

119903)


1198991 119899119903

(1199091 119909

119903) the


infin

sum1198991 119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(70)

for each 119894 = 1 2 119903 where


0= 1 (71)

and g (120572120573)


defined by [14]


=infin

sum119899=0

g (120572120573)

119899(119909 119910) 119905119899


)

(72)


120585119894


119894 centered 120585

119894= 119911

119894and 120585


0 lt 120576119894lt 1003816100381610038161003816119909119894

1003816100381610038161003816minus1

10038161003816100381610038161003816100381610038161003816100381610038161 minus

119903

sum119896=1

119909119896119911119896

1003816100381610038161003816100381610038161003816100381610038161003816 (73)


119863119898

119911119894(1 minus 119909


119903119911119903)minus120572

119911119896

119894

=

119898

2120587119894

∮C120585119894

120585119896

119894(1 minus 119909



119903119911119903)minus120572

(120585119894minus 119911

119894)119898+1

119889120585119894

=

119898

2120587119894


(120585119894+ 119911

119894)119896


119894(120585119894+ 119911


119903119911119903)minus120572

120585119898+1

119894

119889120585119894

=

119898

2120587119894

119911119896

119894(1 minus 119909


119903119911119903)minus120572


(1 + 120585119894119911

119894)119896

120585119898+1

119894

(1 minus

119909119894120585119894


119903119911119903

)

minus120572

119889120585119894

= 119898119911119896

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894


119903119911119903

minus

1

119911119894

)

= 119898119911119896minus119898

119894(1 minus 119909


119903119911119903)minus120572

timesg(120572minus119896)119898

(

119909119894119911119894


119903119911119903

minus1)

(74)


119863119898

119911119894


119903119911119903)minus120572

119911119896119894

= 119863119898

119911119894

119911119896119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903

=infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903)119889119898

119889119911119898119894

11991111989911sdot sdot sdot 119911119899119894+119896

119894sdot sdot sdot 119911119899119903

119903

= 119911119896minus119898119894

infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903

(75)


infin

sum1198991119899119903=0

119866(120572)

1198991 119899119903

(1199091 119909

119903) [119896 + 119899

119894]11989811991111989911sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572

times g(120572minus119896)119898

(119909119894119911119894


119903119911119903

minus1)

(76)


[119899119894+ 119898]

119898= (119899

119894+ 119898) (119899


119894+ 1)

= (119899119894+ 1)

119898

(77)



infin

sum1198991 119899119903=0

(119899119894+ 1)

119898119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(78)



(1 minus 11990911199051minus 119909


119903119905119903119903)minus120572

=infin

sum1198991119899119903=0

119867(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(79)



1198991 119899119903

(1199091 119909

119903) satisfy

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

= (119899119894minus 119894 + 1)119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

119903

sum119894=1

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119894 + 1)

times 119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

(80)

for 119899119894ge 119894 minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


119867(120572+120573)

1198991119899119903

(1199091 119909

119903)

=1198991

sum1198961=0


sum119896119903=0

119867(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119867(120573)

1198961 119896119903

(1199091 119909

119903)

(81)


= 119894 119909119894

= 0 119910119894

= minus119909119894


1198991 119899119903

(1199091 119909

119903)

Remark 28 If

Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)1199081198961

1sdot sdot sdot 119908119896119903

119903

(82)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 and

120582 = (1205821 120582

119903) 120578 = (120578

1 120578

119903)w = (119908

1 119908

119903) then we

have1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572)

1198991minus1198971 119899119903minus119897119903

(1199091 119909

119903)

times 119867(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

(83)

References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











infin

sum1198991 119899119903=0

(119899119894+ 1)

119898119866(120572)

1198991 119899119903

(1199091 119909

119903) 1199111198991

1sdot sdot sdot 119911119899119903

119903


119903119911119903)minus120572


119898(

119909119894119911119894


119903119911119903

minus1)

(78)



(1 minus 11990911199051minus 119909


119903119905119903119903)minus120572

=infin

sum1198991119899119903=0

119867(120572)

1198991 119899119903

(1199091 119909

119903) 1199051198991

1sdot sdot sdot 119905119899119903

119903

(79)



1198991 119899119903

(1199091 119909

119903) satisfy

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

= (119899119894minus 119894 + 1)119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

119903

sum119894=1

119894119909119894

120597

120597119909119894

119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

=119903

sum119894=1

(119899119894minus 119894 + 1)

times 119867(120572)

1198991 119899119894minus1119899119894minus119894+1119899119894+1 119899119903

(1199091 119909

119903)

(80)

for 119899119894ge 119894 minus 1 119894 = 1 2 119903 119899

119895ge 0 119895 = 119894


1198991 119899119903

(1199091 119909


119867(120572+120573)

1198991119899119903

(1199091 119909

119903)

=1198991

sum1198961=0


sum119896119903=0

119867(120572)

1198991minus1198961 119899119903minus119896119903

(1199091 119909

119903)

times 119867(120573)

1198961 119896119903

(1199091 119909

119903)

(81)


= 119894 119909119894

= 0 119910119894

= minus119909119894


1198991 119899119903

(1199091 119909

119903)

Remark 28 If

Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

=[11989911199011]

sum1198961=0

sdot sdot sdot[119899119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572+120573)

1198991minus11990111198961 119899119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)1199081198961

1sdot sdot sdot 119908119896119903

119903

(82)

where 1198861198961 119896119903

= 0 119901119894isin N 119899

119894 120582

119894 120578

119894isin N

0 119894 = 1 2 119903 and

120582 = (1205821 120582

119903) 120578 = (120578

1 120578

119903)w = (119908

1 119908

119903) then we

have1198991

sum1198971=0


sum119897119903=0

[11989711199011]

sum1198961=0

sdot sdot sdot[119897119903119901119903]

sum119896119903=0

1198861198961 119896119903

119867(120572)

1198991minus1198971 119899119903minus119897119903

(1199091 119909

119903)

times 119867(120573)

1198971minus11990111198961 119897119903minus119901119903119896119903

(1199091 119909

119903)

times 119867(120574)

1205821+12057811198961 120582119903+120578119903119896119903

(1199061 119906

119903)

times 1199081198961

1sdot sdot sdot 119908119896119903

119903

= Ξnp120582120578120572120573120574

(1199091 119909

119903 119906

1 119906

119903w)

(83)

References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article On a Family of Multivariate Modified ...e Scientic World Journal bilateral generating functions for the family of multivariable polynomials generated by ( ) and given