In search of higher Bochner theorem

Abstract

We initiate the study of a natural generalisation of the classical Bochner-Krall problem asking which linear ordinary differential operators possess sequences of eigenpolynomials satisfying linear recurrence relations of finite length; the classical case corresponds to the 3-term recurrence relations with real coefficients subject to some extra restrictions. We formulate a general conjecture and prove it in the first non-trivial case of operators of order 3.

Substantially revised and shortened version. Content of Sections 4 and 5 simplified. Computational formulas in Appendix ii corrected. To appear in JAT