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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 419365, 5 pages
Some New Generating Functions for -Hahn Polynomials
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.
We obtain some new generating functions for -Hahn polynomials and give their proofs based on the homogeneous -difference operator.
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 , 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.  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  We have Clearly, .
Recently, Chen et al.  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:
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 ,
Theorem 6. For , , ,
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 ,
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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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