Bivariate second--order linear partial differential equations and orthogonal polynomial solutions

TL;DR

This paper develops algebraic and differential properties of orthogonal polynomial solutions to bivariate second-order linear PDEs, providing explicit formulas, Rodrigues representations, and applications to Appell and Koornwinder polynomials.

Contribution

It introduces a comprehensive framework for orthogonal polynomial solutions of bivariate PDEs, including explicit property formulas and Rodrigues representations, with applications to specific polynomial families.

Findings

Explicit formulas for algebraic and differential properties

Rodrigues representations for polynomial solutions

Application to Appell and Koornwinder polynomials

Abstract

In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially self--adjoint and of hypergeometric type. General formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation, using vector matrix notation. Moreover, Rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given. Finally, as illustration, these results are applied to specific Appell and Koornwinder orthogonal polynomials, solutions of the same partial differential equation.