Partial differential equations having orthogonal polynomial solutions

Abstract

We show that if a second order partial differential equation: L[u] := Au-xx + 2Bu(xy) + Cu-yy + Du(x) + Eu-y = lambda(n)u has orthogonal polynomial solutions, then the differential operator L[.] must be symmetrizable and can not be parabolic in any nonempty open subset of the plane. We also find Rodrigues type formula for orthogonal polynomial solutions of such differential equations. (C) 1998 Elsevier Science B.V. All rights reserved.