Partial Differential Equations and Bivariate Orthogonal Polynomials

In 1929, S. Bochner identified the families of polynomials which are eigenfunctions of a second-order linear differential operator. What is the appropriate generalization of this result to bivariate polynomials? One approach, due to Krall and Sheffer in 1967 and pursued by others, is to determine which linear partial differential operators have orthogonal polynomial solutions with all the polynomials in the family of the same degree sharing the same eigenvalue. In fact, such an operator only determines a multi-dimensional eigenspace associated with each eigenvalue; it does not determine the individual polynomials, even up to a multiplicative constant. In contrast, our approach is to seek pairs of linear differential operators which have joint eigenfunctions that comprise a family of bivariate orthogonal polynomials. This approach entails the addition of some “normalizing" or “regularity" conditions which allow determination of a unique family of orthogonal polynomials. In this article we formulate and solve such a problem and show with the help of Mathematica that the only solutions are disk polynomials. Applications are given to product formulas and hypergroup measure algebras.