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The Cauchy Operator and the Homogeneous Hahn Polynomials

Received: 31 March 2021     Accepted: 23 April 2021     Published: 27 May 2021
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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)).

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.

Copyright

Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

The Cauchy Operator, The Hahn Polynomials, Mehler’s Formula, Rogers Formula

References
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[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.
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[15] Z. Y. Jia, Two q-exponential operator identities and their applications, L. Math. Anal. App. 419 (2014), 329-338.
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  • 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

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    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

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    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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  • @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}
    }
    

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  • 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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Author Information
  • Department of Mathematics, Luoyang Normal University, Luoyang, P. R. China

  • Department of Mathematics, Luoyang Normal University, Luoyang, P. R. China

  • Department of Mathematics, Luoyang Normal University, Luoyang, P. R. China

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