Mellin transforms for several q-polynomials

Bingbing Li Ying Zhang

Bingbing Li , Ying Zhang

1.Anyang Vocational and Technical College , He Nan Province,455000,china

2.No.55 Middle School of Luoyang, He Nan Province, 471000 ,china

Abstract: By Mellin integral transforms for the functionmultiplied byof the type (with an arbitrary constant ,we obtain the Mellin integral transforms for the q-Krawtchouk polynomials, the q-Meixner polynomials.

Key Words: Mellin integral transforms; q-polynomials

1 Introduction

We recall some standard notations for q-shifted factorial and basic hypergeometric series in[1]. For two complex a and q, the shifted factorial with base q is defined by

for ,.

As usual, the basic hypergeometric series is defined by

,.

The Ramanujan's q-extension [2, 3] of the Euler representation for the gamma function is,(1)

Where , .

It was shown in [4] that by using a q-analogue of Euler's reflection formula

is the theta-function of Jacobi, one can represent (1) in the form

(2)

wherefor any nonnegative integer n. Mellin integral transform is defined by. It is easy to see that, when , Mellin integral transform reduces to the Euler representation for the gamma function , When , .

Using (2), we have .

In [5], ,(3)

Whereis a polynomial of degree n in x. When,the Mellin integral transform (3) turns to

,(4)

By, wehave , we have

(5)

Several examples

First, take q-Krawtchouk polynomialsfor example, we have the following Mellin integral transform:

Transformation 1:

(6)

Let (7) substituting (7) in to (4),we obtain the following Mellin integral transform for the q-Krawtchouk polynomials :

Transformation 2:

Similarly, for the q-Meixner polynomials ,we have

Transformation 3:

.

References

[1] G. Gasper, M. Rahman, Basic Hypergeometric Series, second ed., Cambridge University

Press, Cambridge, 2004.

[2] R. Askey, Ramanujans extensions of the gamma and beta functions, Amer. Math.

Monthly 87 (5) (1980) 346-359.

[3] R. Askey, Beta integrals and q-extensions, in: Proceedings of the Ramanujan Centen-

nial International Conference, Annamalainagar, December 15-18, 1987, The Ramanujan

Mathematical Society, Annamalainagar, India, 1988, pp. 85-102.

[5] N.M. Atakishiyev, M.K. Atakishiyeva, A q-analogue of the Euler gamma integral, The-

oret. Math. Phys. 129 (1) (2001) 1325-1334.

[6] S. Ramanujan, Some definite integrals, Messenger Math. 44 (1915) 10-18 (reprinted

in: G.H. Hardy, P.V. Seshu Aiyar, B.M. Wilson (Eds.), Collected Papers of Srinivasa

Ramanujan, Cambridge University Press, Cambridge, 1927, pp.53-58).