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

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. As illustration, these results are applied to a two parameter monic Appell polynomials. Finally, the non-monic case is briefly discussed.