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On some $ 2D$ orthogonal $ q$-polynomials


Authors: Mourad E. H. Ismail and Ruiming Zhang
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
Published electronically: July 7, 2017
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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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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
Email: mourad.eh.ismail@gmail.com

Ruiming Zhang
Affiliation: College of Science, Northwest A&F University, Yangling, Shaanxi 712100, People’s Republic of China
Email: ruimingzhang@yahoo.com

DOI: https://doi.org/10.1090/tran/6824
Keywords: Disc polynomials, Zernike polynomials, 2$D$-Hermite polynomials, $q$-2$D$-Hermite polynomials, generating functions, ladder operators, $q$-Sturm--Liouville equations, $q$-integrals, $q$-Zernike polynomials, Ramanujan's beta integrals, large degree asymptotics, scaled asymptotics, connection relations, Askey--Roy integral, Rogers--Ramanujan identities.
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
Article copyright: © Copyright 2017 American Mathematical Society