Journal of Applied Mathematics

Volume 2014 (2014), Article ID 419365, 5 pages

http://dx.doi.org/10.1155/2014/419365

Research Article

Some New Generating Functions for -Hahn Polynomials

Yun Zhou¹ and Qiu-Ming Luo²

¹Basic Courses Department, Southeast University Chengxian College, Dongda Road, Pukou, Nanjing 210088, China

²Department of Mathematics, Chongqing Higher Education Mega Center, Chongqing Normal University, Huxi Campus, Chongqing 401331, China

Received 10 April 2014; Accepted 25 May 2014; Published 9 June 2014

Academic Editor: Senlin Guo

Copyright © 2014 Yun Zhou and Qiu-Ming Luo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We obtain some new generating functions for -Hahn polynomials and give their proofs based on the homogeneous -difference operator.

1. Introduction

Throughout this paper we suppose that , , and the -shifted factorials are defined by

Clearly,

We also adopt the following compact notation for the multiple -shifted factorials:

The basic hypergeometric series or -series are defined by

Euler identity is as follows:

The -binomial theorem is as follows:

The usual -differential operator or -derivative operator is defined by (see [1, Page 177, (2.1)])

In [1], Chen and Liu introduced the -exponential operator as follows (see [1, Page 17, (2.5)]):

and they get the -operator identity of (see [1, Page 178, Theorems  2.2 and  2.3]) as follows:

Recently Chen et al. [2] introduced the following homogeneous -difference

and the homogeneous -difference operator :

They obtained some properties of as follows:

The classical Rogers-Szegö polynomial is defined by means of the generating function:

obviously, we have

The homogeneous Rogers-Szegö polynomial is defined by

where . Clearly, are the Cauchy polynomials with the following generating function:

From the above properties, we have

Lemma 1 (see [3, Lemma  2.3]). For , ,

-Hahn polynomial is defined by [4]

We have

Clearly, .

Recently, Chen et al. [3] gave some new proofs of the following results based on the method of homogeneous -difference operator .

Theorem 2. Consider the following:

Theorem 3. Consider the following:

For more references on the -difference operators, see [1, 5–16].

In the present paper, we obtain some new generating functions for -Hahn polynomials and give their proofs based on the homogeneous -difference operator.

2. Some New Generating Functions for -Hahn Polynomial

In the present section we obtain the following new generating functions of -Hahn polynomial.

Theorem 4. For ,

Proof. Let and in (21), we have

By the -binomial theorem (6) and noting that , we have

By (17), (25), and (26), we obtain

Comparing the coefficients of on both sides of (27), we obtain the formula (24) immediately. This proof is complete.

Theorem 5. For ,

Proof. By (17) and (19), we have

Setting , , in the last sum, we obtain the formula (28) of Theorem 5. This proof is complete.

Theorem 6. For , , ,

Proof. By (17) and (19), we have

Setting , , in the last sum, we obtain the formula (30) of Theorem 6. This proof is complete.

Theorem 7. For ,

Proof. Applying (2) and the Euler identity (5) and noting (21), then the right-hand side is equal to (30) as follows:

By (30) and (33), we have

Comparing the coefficients of on both sides of (34), we obtain the formula (32) immediately.

Theorem 8. For ,

Proof. Set and then let in (32) and note that ; by (21) and (22), we obtain

This proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The present investigation was supported by the Natural Science Foundation Project of Chongqing, China, under Grant CSTC2011JJA00024, the Research Project of Science and Technology of Chongqing Education Commission, China, under Grant KJ120625, and the Fund of Chongqing Normal University, China, under Grant nos. 10XLR017 and 2011XLZ07.

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