Partial differential equations and bivariate orthogonal polynomials

In 1929, S. Bochner identified the families of polynomials which are eigenf unctions of a second-order linear differential operator. What is the approp riate generalization of this result to bivariate polynomials? One approach, due to Krall and Sheffer in 1967 and pursued by others, is to determine wh ich linear partial differential operators have orthogonal polynomial soluti ons with all the polynomials in the family of the same degree sharing the s ame eigenvalue. In fact, such an operator only determines a multi-dimension al eigenspace associated with each eigenvalue; it does not determine the in dividual polynomials, even up to a multiplicative constant. In contrast, ou r approach is to seek pairs of linear differential operators which have joi nt eigenfunctions that comprise a family of bivariate orthogonal polynomial s. This approach entails the addition of some "normalizing" or "regularity" conditions which allow determination of a unique family of orthogonal poly nomials. In this article we formulate and solve such a problem and show wit h the help of Mathematica that the only solutions are disk polynomials. App lications are given to product formulas and hypergroup measure algebras. (C ) 1999 Academic Press.