On some 2𝐷 orthogonal 𝑞-polynomials

Ismail, Mourad; Zhang, Ruiming

doi:10.1090/tran/6824

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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