Classes of Bivariate Orthogonal Polynomials

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.

References in corpus (2)

• On some 2D orthogonal q-polynomials
• Modular Structures on Trace Class Operators and Applications to Landau Levels

Cited by in corpus (5)

• Quantum superintegrable Zernike system
• New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion
• Superintegrable classical Zernike system
• $\mathcal{D}$-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization
• Interbasis expansions in the Zernike system