The Cauchy operator plays important roles in the theory of basic hypergeometric series. As some applications, our purpose is mainly to show new proofs of the Mehler’s formula, the Rogers formula and the generating function for the homogeneous Hahn polynomials Φ(α)n(x,y|q)) by making use of the Cauchy operator and its properties. In addition, some interesting results are also obtained, which include a formal extension of the generating function for Φ(α)n(x,y|q)).
| Published in | American Journal of Applied Mathematics (Volume 9, Issue 3) |
| DOI | 10.11648/j.ajam.20210903.11 |
| Page(s) | 64-69 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2021. Published by Science Publishing Group |
The Cauchy Operator, The Hahn Polynomials, Mehler’s Formula, Rogers Formula
| [1] | W. A. Al-Salam and L. Carlitz, Some orthogonal q-polynomials, Math. Nachr. 30 (1965), 47-61. |
| [2] | J. Cao, Note on Carlitz’s q-operators, Taiwanese J. Math.14 (2010) 2229-2244. |
| [3] | VY. B. Chen and NS. S. Gu, The Cauchy operator for basic hypergeometric series, Adv. Appl. Math. 41, 177-196 (2008). |
| [4] | VY. B. Chen and Z.-G. Liu, Parameter augmentation for basic hypergeometric series, I, J. Comb. Theory, Ser. A 80(1997), 175-195. |
| [5] | VY. B. Chen and Z.-G. Liu, Parameter augmentation for basic hypergeometric series, II, In: Sagan, BE, Stanley, RP (eds.) Mathematical Essays in honor of Gian-CarloRota, 111-129 (1998). |
| [6] | J.P. Fang, q-differential operator and its applications, J.Math. Anal. Appl. 332 (2007), 1393-1407. |
| [7] | J.P. Fang, Some applications of q-differential operator, J. Korean Math, Soc. 47 (2010), 223-233. |
| [8] | Gasper, G: q-Extension of Barnes’, Cauchy’s and Euler’s beta integrals. In: Rassias, TM (ed.)Topics in Mathematical Analysis, 294-314, (1989). |
| [9] | G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press, Cambridge, MA, 2004. |
| [10] | W. Hahn, Über Orthogonalpolynome, dieq-Differenzengleichungen, Math. Nuchr. 2 (1949) 4-34. |
| [11] | W. Hahn, Beiträgezur Theorieder Heineschen Reihen; Die 24 Integrale der hypergeometrischen q-Differenzengleichung; Das a-Analogon der Laplace-Transformation, Math. Nachr. 2 (1949) 340-379. |
| [12] | Z.-G. Liu, q-Hermite polynomials and a q-beta integral, Northeast. Math. J. 13 (1997) 361-366. |
| [13] | Z.-G. Liu, A q-Extension of a partial differential equation and the Hahn polynomials, Ramanujan J. 38 (2015): 481-501. |
| [14] | ME. H. Ismail, D. Stanton, G. Viennot, The combinatorics of q-Hermite polynomials and the Askey-Wilson integral, Eur. J. Comb. 8 (1987), 379-392. |
| [15] | Z. Y. Jia, Two q-exponential operator identities and their applications, L. Math. Anal. App. 419 (2014), 329-338. |
| [16] | Z. Z. Zhang, J. Wang, Two operator identities and their applications to terminating basic hypergeometric series and q-integrals, J. Math. Anal. Appl. 312 (2005), 653-665. |
APA Style
Qiuxia Hu, Xinhao Huang, Zhizheng Zhang. (2021). The Cauchy Operator and the Homogeneous Hahn Polynomials. American Journal of Applied Mathematics, 9(3), 64-69. https://doi.org/10.11648/j.ajam.20210903.11
ACS Style
Qiuxia Hu; Xinhao Huang; Zhizheng Zhang. The Cauchy Operator and the Homogeneous Hahn Polynomials. Am. J. Appl. Math. 2021, 9(3), 64-69. doi: 10.11648/j.ajam.20210903.11
AMA Style
Qiuxia Hu, Xinhao Huang, Zhizheng Zhang. The Cauchy Operator and the Homogeneous Hahn Polynomials. Am J Appl Math. 2021;9(3):64-69. doi: 10.11648/j.ajam.20210903.11
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Download@article{10.11648/j.ajam.20210903.11,
author = {Qiuxia Hu and Xinhao Huang and Zhizheng Zhang},
title = {The Cauchy Operator and the Homogeneous Hahn Polynomials},
journal = {American Journal of Applied Mathematics},
volume = {9},
number = {3},
pages = {64-69},
doi = {10.11648/j.ajam.20210903.11},
url = {https://doi.org/10.11648/j.ajam.20210903.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20210903.11},
abstract = {The Cauchy operator plays important roles in the theory of basic hypergeometric series. As some applications, our purpose is mainly to show new proofs of the Mehler’s formula, the Rogers formula and the generating function for the homogeneous Hahn polynomials Φ(α)n(x,y|q)) by making use of the Cauchy operator and its properties. In addition, some interesting results are also obtained, which include a formal extension of the generating function for Φ(α)n(x,y|q)).},
year = {2021}
}
TY - JOUR T1 - The Cauchy Operator and the Homogeneous Hahn Polynomials AU - Qiuxia Hu AU - Xinhao Huang AU - Zhizheng Zhang Y1 - 2021/05/27 PY - 2021 N1 - https://doi.org/10.11648/j.ajam.20210903.11 DO - 10.11648/j.ajam.20210903.11 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 64 EP - 69 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20210903.11 AB - The Cauchy operator plays important roles in the theory of basic hypergeometric series. As some applications, our purpose is mainly to show new proofs of the Mehler’s formula, the Rogers formula and the generating function for the homogeneous Hahn polynomials Φ(α)n(x,y|q)) by making use of the Cauchy operator and its properties. In addition, some interesting results are also obtained, which include a formal extension of the generating function for Φ(α)n(x,y|q)). VL - 9 IS - 3 ER -
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