Revista Téc. Ing. Unlv. Zulla. Vol. 19, No. 2 . 93-95, 1996

A note on some integrals involving two associated Laguerre polynomials

Nabil T. Shawagfeh

Mathematics Department. University 01J ordart Amman.JORDAN

Abstract

A c10sed form expresslon In terms ofAppells function F2 is derived for the integral oftwo Associated Laguerre polynomials of the form

00

fe -u x A- t L~(ax) L~(~x)dx O

Sorne Integral formulas correspondlng to special values of the parameters and the arguments are deduced. the results are belleved to be new.

Key worda: Laguerre polynomlals, Appell's functlons, hypergeometric functions.

Una nota sobre algunos integrales que involucran dos polinomios asociados de Laguerre

Resumen

Se deriva una expresión en forma cerrada en términos de la función F2 de AppeU para la Integral de dos polinomios asociados de Laguerre de la forma

-Je- u x A- t L~(ax)L"m(~x)dx o

Se deducen algunas fórmulas Integrales correspondientes a valores especiales de los parámetros y del argumento. Se cree que los resultados son nuevos.

Palabru claves: Polinomios Laguerre, funciones Appell, funciones hlpergeométricas.

Introduction

Thls paper Is concerned wlth the evalu­atlon of the Laplace type Integral

1= Je-SXx)"-IL~(ax)r!'m (~x)dx , (1)

O a where I.:n Is the assoclated Laguerre polynomial. For specUlc values of the parameters, the inte­gral (1) encountered In vartous mathematical physics problems, these Includes, the N-dimen­sional quantum hydrogen atom, and harmonic osclllator [1-21, the electron gas In magnetic fteld

[31. and others [4-61. Sorne of these Integrals are tabulated [7-81.

As far as we know. the most generallnte­gral of the type (1). which has been evaluated In closed form and asymptoticalIy (311s when s =a =P= 1, and thls Integral Is re-evaluated recently [91.

In section 2 the Integral (I) IS evaluated In closed form In terms of Appell's function F2. and In section 3 and through the use of reductlon formulas of F2 for speclal values of the Parame­ters and arguments. sorne Integral formulas are derived.

Rev. Téc. lng. Univ. Zulla. Vol. 19, No. 2. 1996

94 Shawagfeh

Evaluaü on of the Integral I

The associated Lagu erre polynOmial is de­

fmed as

b (b t l)mLm(z)= , l,[-m;b +l,z] , Reb >-) (2)

m.

wherem 1F¡ is the confluent hypergeometric function which is a special case of the generali­zed hypergeometríc function

pF9 [a" a1 ,. •• ,aP ;bl ,b2 ,···, bq;z] = ! (a l »)(a2 )j···(a p ) j j (3) )=0 j! (b¡ )j (b2 ) j ". (bp )j Z

where

na + j) . (a)j= na) =(a)(a+ I) . .. (a +)- 1),(a)0= 1. (4)

The infmite series in (3) terminates If one of the a j •s Is zero or negative Integer. In this work we consider m and n to be non-negative integers and all other parameters to be complexo Thus for the integral (1) to converge we assume that

Re s>O, ReA> O,Re a> -I , Re b>- I . (5)

Substituting

r.!' (~x) = (b+ l)m f (-m). ~. x., (6) m m! k=ok!(b+l)k

in (1) and interchanging the sum and integral we get

1= (b + 1)., ~ (-m). f3 t -J - SI , u ( ) l+'-ltJx (7) ¡ L k '(b 1) e L . ro x ,m. t=O • + • O

which upon using (7,p.844} becomes

(a+ l).(b+ l)~ 1= nlmlsA

"f (-m).rcA+k) ( fJ J' [ a] (8)~ k!(b+l). --; ,F, -n.A+k;a+ l;-;

Expanding the functlons 2Fl using (3) and making use of the identlty

_ (9)r~(A~)(_A):...:...h:..!.j(A+k) =

) f(A + k)

we obtain (a + l).(b + I)J(A)

1= .n!m!s Á

(lO)

The double series in (lO) is Appell's functlon

F2 defined as [lO)

.. .. (A) ·(B ) (B' ) · F [A B B'·C C'· 1= ~ ~ *t] k ] k jo , , , , ,x, y LL x y (11)

- . k;,lI j =O klj!(Cl d C ) j

which converges for Ixl +Iyl < 1 . Thus, we obtain the main formula

(a+l )n (b+l)mfO..) [ ex ~]1= F¿ A,-n,-m;a+l,b+ I;-,- . (12) n!m l / s s

Special Cases

Appell's functlon F2 reduces to simpler functions for speclal values of its parameters and arguments. In the following we consider two special cases which lead to new integral formu­las.

Case s =a

When one of the arguments of Appell's function F2is un ity it reduces to 3F2 funcUon, In particular, It is easy to show that

F [A- B B" C C 'I 1= rcC)r(c - A - B ) , ' , , , ..Y rcC-A)f(C-B) '

r A, B' ,1 + A .. C 1 3F2lc. I+A+B_C;YJ (13)

hence, when s=a in (12) we obtain

(14)

provided that I~ < laI and Re(a) > O.

Case s=a+~

One of the transformaUon of the F 2 func­Uons 1s (10]

F, [A;B .B';C,C;x. ,r] ~ (1- yr A F¡[A;B;C-B';C,C;_x- .-)-' ](5)1- Y )'-1

wherel.x1+lYI < II - yl , wh1ch gtves the alternate formula

1= (a+ I).(b+ 1) ", f(A)

ni m! (s- fJ )A

r a fJ]F'lA.;-n,m+b+ I;a+ I,b+ 1;--,-- (16)s-fJ fJ-s

Rev. Téc. Ing. Univ. Zulia. Vol. 19. No. 2. 1996

95 Integrals involVing Laguerre polynonúals

Substltuttng s=a+~ in (16) and using (13) we obtain

fe- (a+B)xx A-I L~ (ax)L~ (Jh)dx= O

(b+l ) (a-A+l ) nA) [A ,b +m+I.A-a P]m n )FZ ;-- (1 7)n ! m! a). b+1.A-a-n a

provided that IPI <lal and Re(a + ~) > O.

Using the syrnmetry of the integral. ana­logue formulae for (1 4) and (1 7) can be derived for the case I~I > lal by interchanging at--'f b, m~ n and a ~.B . Furthermore. several int­

egral formulae can be deduced as speclal cases

coí [14) Md (1 7} usin5 the reducUon fmnwla; gf 3F2 function.

Conclusion

We have derived a closed form expression for the integraJ (1) from which we have deduced two integral formulas for special values of the parameters. these formu las are believed to be new.

References

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Recibido el 23 de Marzo de 1995

En fonna revisada el 22 de Enero de 1996

Rev. Téc. Ing. Univ. Zulla. Vol. 19. No. 2 . 1996