Second order difference equations and discrete orthogonal polynomials of two variables

Abstract

The second-order partial difference equation of two variables Du:=A1,1(x)Δ1∇1u+A1,2(x)Δ1∇2u+A2,1(x)Δ2∇1u+A2,2(x)Δ2∇2u+B1(x)Δ1u+B2(x)Δ2u=λu is studied to determine when it has orthogonal polynomials as solutions. We derive conditions on D so that a weight function W exists for which W D u is selfadjoint 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.