An extension of Bochner's problem: exceptional invariant subspaces

Abstract

We prove an extension of Bochner's classical result that characterizes the classical polynomial families as eigenfunctions of a second-order differential operator with polynomial coefficients. The extended result involves considering differential operators with rational coefficients and the requirement is that they have a numerable sequence of polynomial eigenfunctions p(1), p(2),.. of all degrees except for degree zero. The main theorem of the paper provides a characterization of all such differential operators. The existence of such differential operators and polynomial sequences is based on the concept of exceptional polynomial subspaces, and the converse part of the main theorem rests on the classification of codimension one exceptional subspaces under projective transformations, which is performed in this paper.