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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On some $2D$ orthogonal $q$-polynomials
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by Mourad E. H. Ismail and Ruiming Zhang PDF
Trans. Amer. Math. Soc. 369 (2017), 6779-6821 Request permission

Abstract:

We introduce two $q$-analogues of the $2D$-Hermite polynomials which are functions of two complex variables. We derive explicit formulas, orthogonality relations, raising and lowering operator relations, generating functions, and Rodrigues formulas for both families. We also introduce a $q$-$2D$ analogue of the disk polynomials (Zernike polynomials) and derive similar formulas for them as well, including evaluating certain connection coefficients. Some of the generating functions may be related to Rogers–Ramanujan type identities.
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Additional Information
  • Mourad E. H. Ismail
  • Affiliation: College of Science, Northwest A&F University, Yangling, Shaanxi 712100, People’s Republic of China – and – Department of Mathematics, University of Central Florida, Orlando, Florida 32816
  • MR Author ID: 91855
  • Email: mourad.eh.ismail@gmail.com
  • Ruiming Zhang
  • Affiliation: College of Science, Northwest A&F University, Yangling, Shaanxi 712100, People’s Republic of China
  • MR Author ID: 257230
  • Email: ruimingzhang@yahoo.com
  • Received by editor(s): November 5, 2014
  • Received by editor(s) in revised form: June 16, 2015, and August 14, 2015
  • Published electronically: July 7, 2017
  • Additional Notes: The research of the first author was supported by the DSFP at King Saud University in Riyadh, by Research Grants Council of Hong Kong contract #1014111, and by the National Plan for Science, Technology and innovation (MAARIFAH), King Abdelaziz City for Science and Technology, Kingdom of Saudi Arabia, Award No. 14-MAT623-0
    The research of the second author was supported by Research Grants Council of Hong Kong, Contract #1014111, and the National Science Foundation of China, grant No. 11371294. The second author is the corresponding author
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 6779-6821
  • MSC (2010): Primary 33C50, 33D50; Secondary 33C45, 33D45
  • DOI: https://doi.org/10.1090/tran/6824
  • MathSciNet review: 3683094