On polynomial solutions of certain finite order ordinary differential equations

Abstract

Given a finite order differential operator, some properties and relations satisfied by its polynomial eigenfunctions are studied. Under certain restrictions, such eigenfunctions are explicitly obtained, as well as the corresponding eigenvalues. Also, some linear transformations are applied to sequences of eigenfunctions and a necessary condition for this to be a sequence of eigenfunctions of a new differential operator is obtained. These results are applied to the particular case of classical Hermite polynomials.