Second order difference equations and discrete orthogonal polynomials of two variables

TL;DR

This paper investigates second order partial difference equations in two variables to identify conditions under which they admit orthogonal polynomial solutions, linking to classical discrete orthogonal polynomials.

Contribution

It derives conditions for the existence of weight functions making the difference operator self-adjoint, leading to orthogonal polynomial solutions in two variables.

Findings

Conditions for orthogonal polynomial solutions are established.

Classical discrete orthogonal polynomials are characterized as solutions.

Self-adjointness of the difference operator is linked to weight functions.

Abstract

The second order partial difference equation of two variables $ \CD u:= A_{1,1}(x) \Delta_1 \nabla_1 u + A_{1,2}(x) \Delta_1 \nabla_2 u + A_{2,1}(x) \Delta_2 \nabla_1 u + A_{2,2}(x) \Delta_2 \nabla_2 u & \qquad \qquad \qquad \qquad + B_1(x) \Delta_1 u + B_2(x) \Delta_2 u = \lambda u, $ is studied to determine when it has orthogonal polynomials as solutions. We derive conditions on $\CD$ so that a weight function $W$ exists for which $W \CD u$ is self-adjoint and the difference equation has polynomial solutions which are orthogonal with respect to $W$. The solutions are essentially the classical discrete orthogonal polynomials of two variables.