Classes of Bivariate Orthogonal Polynomials

[摘要] 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.

[关键词] disc polynomials;Zernike polynomials;2D-Laguerre polynomials;q-2D-Laguerre polynomials;generating functions;ladder operators;q-Sturm–Liouville equations;q-integrals;q-Zernike polynomials;2D-Jacobi polynomials;q-2D-Jacobi polynomials;connection relations;biorthogonal functions;generating functions;Rodrigues formulas;zeros