Matrix Pearson equations satisfied by Koornwinder weights in two variables

TL;DR

This paper explores how Koornwinder's method constructs two-variable orthogonal polynomials from one-variable cases, deriving matrix Pearson equations and differential operators for classical Koornwinder polynomials.

Contribution

It introduces two methods to derive matrix Pearson equations for weights associated with Koornwinder polynomials, extending classical results to two variables.

Findings

Derived matrix Pearson equations for Koornwinder weights.

Established second order linear PDEs for classical Koornwinder polynomials.

Connected semiclassical properties in one and two variables.

Abstract

We consider Koornwinder's method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder's construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with these polynomials, and consequently, we deduce the second order linear partial differential operators for classical Koornwinder polynomials.