Classes of Bivariate Orthogonal Polynomials

TL;DR

This paper introduces a new class of bivariate orthogonal polynomials that generalize existing polynomials like disc and Hermite polynomials, providing their properties, extensions, and $q$-analogues.

Contribution

It defines a broad class of bivariate orthogonal polynomials, including notable special cases and their $q$-analogues, with detailed properties and operator characterizations.

Findings

Identification of interesting polynomial subclasses

Development of generating functions and recursion relations

Establishment of differential and $q$-difference operator eigenfunctions

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.