Global Journal of Pure and Applied Mathematics.

ISSN 0973-1768 Volume 16, Number 1 (2020), pp. 119-130

© Research India Publications

http://www.ripublication.com/gjpam.htm

Some Results Associated with Jacobi Matrix

Polynomials

Mosaed M. Makky

Department of Mathematics, Faculty of Science, South Valley University(Qena-Egypt)

Abstract

In this paper the main idea is to obtain some recurrence relations and generating

matrix function for Jacobi matrix polynomials of two complex variables by acting

of differential operator.

Keywords: Jacobi matrix polynomials, differential operator, recurrence relations,

hypergeometric function.

2010 Mathematics Subject Classification: Primary 33C25 Secondary 15A60.

1. INTRODUCTION

The study of the functions of special matrix observed in previous studies and both the

Lie group theory, number theory are known in [1, 2]. Also in recent studies, a

polynomial matrix appeared in [3, 4, 5, 6]. In [7, 8, 9, 10, 11, 12] the authors

presented and study the Jacobi matrix polynomials.

The theory of orthogonal polynomials extends to a polynomials matrix, as in papers

[13, 14, 15].

Some results in classical orthogonal polynomials theory extended to orthogonal

polynomials in [4, 16].

Our main objective in this paper is to demonstrate new polynomial properties of the

Jacobi matrix under to use of some differential operators.

Throughout for a matrix NxNA , its spectrum ( )A denotes the set of all

eigenvalues of matrix A.

If f (z) and g(z) are holomorphic functions of the one complex variable z, which are

defined in an open set of the complex plane and ,A B are matrices in NxN with

120 Mosaed M. Makky

( )A and ( )B ; then from the properties of the matrix functional calculus

in [17], it follows that

( ) ( ) ( ) ( )f A g B g B f A . (1.1)

If NxNA is a matrix such that A nI is an invertible matrix for all integer 0n

we have the matrix version of the pochhammer symbol is

( ) ( )( 2 ) ... ( ( 1) ) ; 1nA A A I A I A n I n ; 0( )A I . (1.2)

The classical Jacobi matrix polynomials ( , ) ( )A BnP x of one variable as given

( , )

2 1

( ) 1( ) , ; ;

! 2

A B nn

I A zP z F nI I A B nI I An

(1.3)

where the hypergeometric function write as follows c.f.[1]):

1

2 1

0

( , )( , )( , )( ; , ; )

(1, )

n

n

A n B n C nF A B C z zn

for matrices , , NxNA B C such that C nI is invertible for all integer 0n and

for |z| < 1.

For A is an arbitrary matrix in NxN and using (1.2), we have the following relations

(c.f. [18])

( ) ( ) ( )n k n k k nA A A nI A A kI

2 2 2( ) ( ) ( 2 )n k n k k nA A A nI A A kI

and

2

2

1 1( ) 2 ( )

2 2

kk

k k

A A A I

.

Cekim B. and etc. (c.f. [19]) some recurrence relations for Jacobi matrix polynomials

(JMP) are given and a generating matrix function for JMP is also obtained in this

article furthermore, we show the integral representations for JMP are given it, for A

and B be matrices in NxN whose eigenvalues, z, all satisfy Re( ) 1z , Jacobi

matrix polynomials satisfies some recurrence relations which satisfied by Jacobi

matrix polynomials (JMP) of one variable (c.f. [19]) are given in the following:

Theorem (1.1)

Let A and B be matrices in NxN whose eigenvalues z, all satisfy Re( ) 1z (JMP)

then we have

i. ( , ) ( , )

1

0

1( ) ( )

2

nA B A I B I

n nk

n I A Bd P x P xdx

Some Results Associated with Jacobi Matrix Polynomials 121

ii.

( , ) ( , )1

( ) ( )2

sA B A sI B sIs

n n ss s

n I A Bd P x P xdx

for 0 s n

iii. ( , ) ( , )( ) ( 1) ( )A B n A Bn nP x P x .

Theorem (1.2)

Let A and B be matrices in NxN whose eigenvalues z, all satisfy Re( ) 1z . For

the Jacobi matrix polynomials (JMP), the following recurrence relations are achieved:

( , ) ( , ) ( , )

1( 1) ( ) ( ) ( )A B A B A I Bn n n

dx P x n P x B nI P xdx

and

( , ) ( , ) ( , )

1( 1) ( ) ( ) ( )A B A B A B In n n

dx P x n P x A nI P xdx

.

From above theorem we can write

( , ) ( , ) ( , )

1 12 ( ) ( ) ( )A B A I B A B In n n

d P x B nI P x A nI P xdx

and

( , ) ( , ) ( , )( 2) ( ) ( 1) ( ) ( 1) ( ).A I B I A I B A B In n nA B n I P x B n I P x A n I P x

In addition to many other results studied in article [20].

The Jacobi matrix polynomials of two complex variables ( , ) ( , )A BnP z w is defined as

follows:

1 1

( , )

0

( , )!( )! 2

kn

A B n n k k nn

k

I A I A B I A I A B z wP z wk n k

where ,A B be a positive stable matrices in NxN , , ( 1,1)z w , ( A B B A ) which

can be simplified to obtain the generating matrix polynomials as follows:

1 1

( , )

0

1 1( , )

!( )! 2 1

kn knA B n n k k n

nk

I B I A B I B I A B z wP z wk n k w

1 1

( , )

, 0

( , )!( )! 2 2

k n k

A B n n k n kn

n k

I A I B I A I B z w z wP z wk n k

122 Mosaed M. Makky

( , )

2 1( , ) , , ;! 2

n

A B nn

I A z w z wP z w F nI B nI I An z w

and

( , )

2 1( , ) , , ;! 2

A B nn

I A w zP z w F nI I A B nI I An

.

2- MAIN RESULTS

In this section we will study the effect of the differential operator on the Jacobi matrix

polynomials to identify the results that can be obtained for this study.

For this propose, let NxNA be a matrix version of the pochhammer symbol is

( ) ( )( 2 )... ( ( 1) ) ; 1nA A A I A I A n I n ; 0( )A I .

Now, we write the conjugate relations that we can use in the following theorem as

follows:

1

2 ( ) ( )n n

I A B I A B I A B I I A I B

(2.1)

1

( )n n

I A I A I I A

(2.2)

1

( )k k

I A I A I I A

(2.3)

1 ( 1)

2 ( ) ( )n k n k

I A B I A B I A B I I A I B

. (2.4)

This is what we will introduced in the following theorem:

Theorem (2.1):

Suppose that the Jacobi matrix polynomials of two complex variables given as

follows:

1 1

( , )

0

( , )!( )! 2

kn

A B n n k k nn

k

I A I A B I A I A B z wP z wk n k

(2.5)

where ,A B be a positive stable matrices in NxN , , ( 1,1)z w , and given the

differential operator wz w

.

Then Jacobi matrix polynomials of two complex variables satisfies the matrix

differential equation:

, ,

1

1( , ) ( , ) 0

4

A B A I B In nP z w P z w

. (2.6)

Some Results Associated with Jacobi Matrix Polynomials 123

Proof:

By the operator wz w

and using the relations (2.1), (2.2), (2.3) and

(2.4) for the Jacobi matrix polynomials of two complex variables ( , ) ( , )nP z w we see

that

( , ) ( , )A Bnw P z w

z w

1 1

0 !( )! 2

kn

n n kk n

k

I A I A B z ww I A I A Bz w k n k

1

1 1

0

1

1 1

0

1 1

!( )! 2 2 4

1

4 !( )! 2

kn

n n kk n

k

kn

n n kk n

k

k I A I A B z wI A I A Bk n k

k I A I A B z wI A I A Bk n k

1 11

11

1

4 !( ( 1))! 2

kn

n n kk n

k

I A I A B z wI A I A Bk n k

.

Now write this relation as follows:

( , ) 1

1 1

1( , )

4 !( ( 1))! 2

kn

A B n n kn

k k n

I A I A B z wP z wk n k I A I A B

1 ( 1)

1

1

( ) 2 ( ) ( )

!( ( 1))! ( )1

4 1

2 ( ) ( ) 2

n n k

nk

kk

n

I A I I A I A B I A B I I A I B

k n k I A I I A

z wI A B I A B I I A I B

1 ( 1)

1 1

( ) ( ) ( )1

4 !( ( 1))! ( ) ( ) ( ) 2

kn

n n k

k k n

I I A I I A I B z wk n k I I A I I A I B

1

1 ( 1)

1 1 1

( ) ( ) ( )

! ( 1) !1

4( ) ( ) ( )

2n

n n k

n

kk

k

I I A I I A I B

k n k

z wI I A I I A I B

, ,

1

1( , ) ( , )

4

A B A I B In nP z w P z w

.

124 Mosaed M. Makky

Then Jacobi matrix polynomials (2.5) satisfies the matrix differential equation:

, ,

1

1( , ) ( , ) 0

4

A B A I B In nP z w P z w

. (2.7)

In the same way as the differential operator on the Jacobi matrix polynomials and

using differential operator suitable on different images of the Jacobi matrix

polynomials we get the similar results:

1 1

( , )

0

( , )!( )! 2 1

kn

A B n n k k nn

k

I A I A B I A I A B z wP z wk n k w

1 1

( , )

0

1 1( , )

!( )! 2 1 2 1

k n kn

A B n n k n kn

k

I A I B I A I B z w z wP z wk n k w w

and

1 1

( , )

0

1 1( , ) .

!( )! 2 1

kn knA B n n k k n

nk

I B I A B I B I A B z wP z wk n k w

RECURRENCE RELATIONS FOR JACOBI MATRIX POLYNOMIALS

Some recurrence relations have been deduced for the Jacobi matrix polynomials, as

follows:

1- At first let ,A B be matrices in N x N where 0 ( ) 1 , for all ( )A ,

( )B and put w=1 in the Jacobi matrix polynomials (2.5) we get

1 1

( , )

0

1( ,1)

!( )! 2

knA B n n k k n

nk

I A I A B I A I A B zP zk n k

where ( 1,1)z .

Now we write the differential operator

zddz

.

Some Results Associated with Jacobi Matrix Polynomials 125

Therefore

( , ) ( ,1)A Bn

d P zdz

1 1

0

1

!( )! 2

knn n k

k nk

I A I A Bd zI A I A Bdz k n k

11 1

0

11 1

0

1 1

!( )! 2 2

1 1

2 !( )! 2

knn n k

k nk

knn n k

k nk

k I A I A B zI A I A Bk n k

k I A I A B zI A I A Bk n k

1 11

11

1 1

2 !( ( 1))! 2

knn n k

k nk

I A I A B zI A I A Bk n k

.

Now write this relation as follows:

( , ) 1

1 1

1 1( ,1)

2 !( ( 1))! 2

knA B n n k

z nk k n

I A I A B zP zk n k I A I A B

1 ( 1)

1 1

( ) ( ) ( )1 1

2 !( ( 1))! ( ) ( ) ( ) 2

knn n k

k k n

I I A I I A I B zk n k I I A I I A I B

1

1 ( 1)

1 1 1

( ) ( ) ( )

! ( 1) !1

2 1( ) ( ) ( )

2n

n n k

n

kk

k

I I A I I A I B

k n k

zI I A I I A I B

, 1, 1

1

1( ,1) ( ,1)

2

A B A Bz n nP z P z

.

Then Jacobi matrix polynomials satisfies the matrix differential equation:

, ,

1

1( ,1) ( ,1) 0

2

A B A I B Iz n nP z P z

. (2.8)

Now by putting w=1 in the different images of the Jacobi matrix polynomials and

with appropriate differential operators we get similar results:

1 1

( , )

0

2( ,1)

!( )! 2 2

kn

A B n n k k nn

k

I A I A B I A I A B zP zk n k

,

1 1

( , )

0

2 2( ,1)

!( )! 2 2 2 2

k n kn

A B n n k n kn

k

I A I B I A I B z zP zk n k

126 Mosaed M. Makky

and

1 1

( , )

0

1 2( ,1)

!( )! 2 2

kn knA B n n k k n

nk

I B I A B I B I A B zP zk n k

.

2- In the same way the following results can be obtained by put z=1 in (2.5) and using

the same operator we get:

1 1

( , )

0

1(1, )

!( )! 2

kn

A B n n k k nn

k

I A I A B I A I A B wP wk n k

where ( 1,1)w .

Now we write the differential operator

w ww

.

Therefore

( , ) (1, )A Bnw P w

w

1 1

0

1

!( )! 2

kn

n n kk n

k

I A I A B ww I A I A Bw k n k

1

1 1

0

1 1

2 !( )! 2

kn

n n kk n

k

k I A I A B wI A I A Bk n k

1 1

1 1

1

1 1

2 !( ( 1))! 2

kn

n n k k n

k

I A I A B I A I A B wk n k

.

Now write this relation as follows:

( , ) 1

1 1

1 1(1, )

2 !( ( 1))! 2

kn

A B n n kw n

k k n

I A I A B wP wk n k I A I A B

1 ( 1)

1 1

( ) ( ) ( )1 1

2 ! ( 1) ! ( ) ( ) ( ) 2

kn

n n k

k k n

I I A I I A I B wk n k I I A I I A I B

1 ( 1)

1 1

( ) ( ) ( )1 1

2 ! ( 1) ! ( ) ( ) ( ) 2

kn

n n k

k k n

I I A I I A I B wk n k I I A I I A I B

Some Results Associated with Jacobi Matrix Polynomials 127

therefore

, ,

1

1(1, ) (1, )

2

A B A I B Iw n nP w P w

. (2.9)

Then Jacobi matrix polynomials satisfies the matrix differential equation:

, ,

1

1(1, ) (1, ) 0

2

A B A I B Iw n nP w P w

.

Again by putting z=1 in the different images of the Jacobi matrix polynomials and

with appropriate differential operators we get similar results for the case w=1.

3- Now by putting A=I in the Jacobi matrix polynomials and using differential

operator wz w

we get:

( , ) ( , )I Bnw P z w

z w

1 1

0

2 22 2

!( )! 2

kn

n n kk n

k

I I B z ww I I Bz w k n k

1

1 1

0

2 212 2

4 !( )! 2

kn

n n kk n

k

k I I B z wI I Bk n k

1 11

11

2 212 2

4 !( ( 1))! 2

kn

n n kk n

k

I I B z wI I Bk n k

.

Now write this relation as follows:

( , ) 1

1 1

2 21( , )

4 ! ( 1) ! 2 2 2

kn

I B n n kn

k k n

I I B z wP z wk n k I I B

1 ( 1)

1 1

3 3 ( )1

4 !( ( 1))! 3 3 ( ) 2

kn

n n k

k k n

I I I B z wk n k I I I B

1

11

1 ( 1)

1

3 3 ( ) 3 3 ( )1

4 ! ( 1) ! 2

n

kn

n kn k

k

I I I B I I I B z wk n k

, 2 ,

1

1( , ) ( , )

4

I B I B In nP z w P z w

.

128 Mosaed M. Makky

Then Jacobi matrix polynomials satisfies the matrix differential equation:

, 2 ,

1

1( , ) ( , ) 0

4

I B I B In nP z w P z w

. (2.10)

Also, by putting A=I in the different images of the Jacobi matrix polynomials and

with appropriate differential operators we get similar results as follows:

1 1

( , )

0

2 2 2 2( , )

!( )! 2 1

kn

I B n n k k nn

k

I I B I I B z wP z wk n k w

1 1

( , )

0

2 2 1 1( , )

!( )! 2 1 2 1

k n kn

I B n n k n kn

k

I I B I I B z w z wP z wk n k w w

and

1 1

( , )

0

1 2 2 1( , ) .

!( )! 2 1

kn knI B n n k k n

nk

I B I B I B I B z wP z wk n k w

4- Now by putting B=I in the Jacobi matrix polynomials and with differential operator

wz w

we get:

( , ) ( , )A Inw P z w

z w

1 1

0

22

!( )! 2

kn

n n kk n

k

I A I A z ww I A I Az w k n k

1

1 1

0

212

4 !( )! 2

kn

n n kk n

k

k I A I A z wI A I Ak n k

1 11

11

212

4 !( ( 1))! 2

kn

n n kk n

k

I A I A z wI A I Ak n k

.

Some Results Associated with Jacobi Matrix Polynomials 129

Now write this relation as follows:

1 ( 1),

1 1

( ) 3 ( )1( , )

4 !( ( 1))! ( ) 3 ( ) 2

kn

n n kA In

k k n

I I A I I A z wP z wk n k I I A I I A

1

1 1

1 ( 1)

1

( ) 3 ( ) ( ) 3 ( )1

4 ! ( 1) ! 2

n

kn

n n k k

k

I I A I I A I I A I I A z wk n k

1

1 1

1 ( 1)

1

( ) 3 ( ) ( ) 3 ( )1

4 ! ( 1) ! 2

n

kn

n n k k

k

I I A I I A I I A I I A z wk n k

i.e.

, ,2

1

1( , ) ( , )

4

A I A I In nP z w P z w

.

Then Jacobi matrix polynomials satisfies the matrix differential equation:

, ,2

1

1( , ) ( , ) 0

4

A I A I In nP z w P z w

. (2.11)

Also, by putting B=I in the different images of the Jacobi matrix polynomials and

with appropriate differential operators we get similar results for the case A=I.

CONCLUSIONS

In this paper a new comparison has been introduced for studying some important

properties of certain matrix special functions matrix recurrence relations, matrix

differential recurrence relations and matrix differential equation. The method

developed in this paper can also be used to study some other special matrix functions

which play a needful role in Mathematical Physics.

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