Families of bivariate orthogonal polynomials that determine differential operators
The classical families of orthogonal polynomials arise as eigenfunctions of Sturm-Liouville problems. In 1929, Bochner addressed the converse question: Which linear second order differential operators can have an infinite family of polynomials, P, as their eigenvalues? The classification that we gave showed that there were, up to a linear change of variables, a unique differential operator associated with each such family, and the argument did not even require that the members of P were orthogonal, only that there were "enough" polynomials in the family (indeed, in some cases, the polynomials are not orthogonal). In this paper it will be shown that families of bivariate polynomials which have certain type of product formula satisfy not just one, but a pair L^(1) and L^(2) of partial differential operators, with bounds on the order of the operators determined by properties of P.