Partial differential equations satisfied by polynomials which have a product formula. (Q1415039)

The paper under review concerns a problem addressed by Bochner in 1929: which linear second-order differential operators can have an infinite family of polynomials \({\mathcal P}\) as their eigenfunctions. His answer was that, as soon as such a family \({\mathcal P}\) contains ``enough'' polynomials, there is a unique differential operator associated to the family \({\mathcal P}\). In this paper it is shown that certain families of bivariate polynomials, which satisfy a product formula on an open set, are eigenfunctions for a pair \(L^{(1)}\), \(L^{(2)}\) of commuting linear partial differential operators. A simple condition that \((L^{(1)}, L^{(2)})\) determines \({\mathcal P}\) uniquely is given, in terms of the corresponding eigenfunctions. Let \(D({\mathcal P})\) be the algebra of linear partial differential operators with polynomial coefficients which have all members of \(\mathcal P\) as eigenfunctions. One gives a sufficient condition that a pair of operators \((L^{(1)}, L^{(2)})\) as above generates \(D({\mathcal P})\). The paper includes detailed discussions of five examples of polynomial families as above, and also four open problems.