e-mail: schwartz@arch.umsl.edu

**Families of bivariate orthogonal polynomials that determine
differential operators**

**Abstract**

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.*