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Journal of Applied Mathematics

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

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

## Some New Generating Functions for -Hahn Polynomials

^{1}Basic Courses Department, Southeast University Chengxian College, Dongda Road, Pukou, Nanjing 210088, China^{2}Department 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.

#### References

- W. Y. C. Chen and Z.-G. Liu, “Parameter augmentation for basic hypergeometric series. II,”
*Journal of Combinatorial Theory. Series A*, vol. 80, no. 2, pp. 175–195, 1997. View at Publisher · View at Google Scholar · View at MathSciNet - W. Y. C. Chen, A. M. Fu, and B. Zhang, “The homogeneous $q$-difference operator,”
*Advances in Applied Mathematics*, vol. 31, no. 4, pp. 659–668, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - W. Y. C. Chen, H. L. Saad, and L. H. Sun, “The bivariate Rogers-Szegö polynomials,”
*Journal of Physics A: Mathematical and Theoretical*, vol. 40, no. 23, pp. 6071–6084, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Hahn, “Über Polynome, die gleichzeitig zwei verschiedenen Orthogonalsystemen angehören,”
*Mathematische Nachrichten*, vol. 2, pp. 263–278, 1949. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Bowman, “$q$-difference operators, orthogonal polynomials, and symmetric expansions,”
*Memoirs of the American Mathematical Society*, vol. 159, no. 757, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Cao, “A note on moment integrals and some applications,”
*Journal of Mathematical Analysis and Applications*, vol. 410, no. 1, pp. 348–360, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Cao, “A note on $q$-integrals and certain generating functions,”
*Studies in Applied Mathematics*, vol. 131, no. 2, pp. 105–118, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - J. Cao, “A note on generating functions for Rogers-Szegö polynomials,”
*Quaestiones Mathematicae*, vol. 35, no. 4, pp. 447–461, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Cao, “Bivariate generating functions for Rogers-Szegö polynomials,”
*Applied Mathematics and Computation*, vol. 217, no. 5, pp. 2209–2216, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Cao, “New proofs of generating functions for Rogers-Szegö polynomials,”
*Applied Mathematics and Computation*, vol. 207, no. 2, pp. 486–492, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Y. B. Chen and N. S. S. Gu, “The Cauchy operator for basic hypergeometric series,”
*Advances in Applied Mathematics*, vol. 41, no. 2, pp. 177–196, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - W. Y. C. Chen and Z.-G. Liu, “Parameter augmentation for basic hypergeometric series. I,” in
*Mathematical Essays in Honor of Gian-Carlo Rota*, B. E. Sagan and R. P. Stanley, Eds., vol. 161, pp. 111–129, Birkhäuser, Boston, Mass, USA, 1998. View at MathSciNet - J.-P. Fang, “$q$-differential operator identities and applications,”
*Journal of Mathematical Analysis and Applications*, vol. 332, no. 2, pp. 1393–1407, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-G. Liu, “Some operator identities and $q$-series transformation formulas,”
*Discrete Mathematics*, vol. 265, no. 1–3, pp. 119–139, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - L. J. Rogers, “On the expansion of some infinite products,”
*Proceedings of the London Mathematical Society*, vol. 24, no. 1, pp. 337–352, 1893. View at Publisher · View at Google Scholar · View at MathSciNet - H. L. Saad and A. A. Sukhi, “Another homogeneous $q$-difference operator,”
*Applied Mathematics and Computation*, vol. 215, no. 12, pp. 4332–4339, 2010. View at Publisher · View at Google Scholar · View at MathSciNet