On polynomial solutions of certain finite order ordinary differential equations

Some properties and relations satisfied by the polynomial solutions of a bispectral problem are studied. Given a finite order differential operator, under certain restrictions, its polynomial 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.