Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 021, 37 pages      arXiv:1502.07256      https://doi.org/10.3842/SIGMA.2016.021
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Classes of Bivariate Orthogonal Polynomials

Mourad E.H. Ismail ab and Ruiming Zhang c
a) Department of Mathematics, King Saud University, Riyadh, Saudi Arabia
b) Department of Mathematics, University of Central Florida, Orlando, Florida 32816, USA
c) College of Science, Northwest A&F University, Yangling, Shaanxi 712100, P.R. China

Received August 04, 2015, in final form February 15, 2016; Published online February 24, 2016

Abstract
We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-$D$ Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the 2-$D$ Hermite polynomials and a two variable extension of the Zernike or disc polynomials. We also give $q$-analogues of all these extensions. In each case in addition to generating functions and three term recursions we provide raising and lowering operators and show that the polynomials are eigenfunctions of second-order partial differential or $q$-difference operators.

Key words: disc polynomials; Zernike polynomials; 2$D$-Laguerre polynomials; $q$-2$D$-Laguerre polynomials; generating functions; ladder operators; $q$-Sturm-Liouville equations; $q$-integrals; $q$-Zernike polynomials; 2$D$-Jacobi polynomials; $q$-2$D$-Jacobi polynomials; connection relations; biorthogonal functions; generating functions; Rodrigues formulas; zeros.

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