Characterization of classical orthogonal polynomials in two continuous variables

TL;DR

This paper characterizes classical orthogonal polynomials in two variables, establishing equivalences among key properties and introducing a new notion of classicality with illustrative examples.

Contribution

It provides a comprehensive characterization of classical orthogonal polynomials in two variables, linking weight functions, differential equations, and structural relations.

Findings

Proves equivalence of properties for orthogonal polynomials in two variables.

Introduces a new notion of classical orthogonal polynomials in two variables.

Provides a nontrivial example illustrating the theoretical framework.

Abstract

For a family of polynomials in two continuous variables, orthogonal with respect to a weight function, we prove, under suitable conditions, the equivalence of the following properties: the matrix Pearson equation of the weight, the second order linear partial differential equation, the orthogonality of the gradients, the matrix Rodrigues formula involving tensor products of matrices, and the so-called first structure relation. We then introduce a notion of classical orthogonal polynomials in two variables and relate the corresponding theory for weight functions and moment functionals. Finally, we present a nontrivial example that illustrates and delineates our contribution to the field.