Polynomial eigenfunctions of linear ordinary differential operators

We study conditions for linear ordinary differential operators with polynomial coefficients to have at least a number of polynomial eigenfunctions. These conditions include the case of infinite sequences of polynomial eigenfunctions (Bochner’s theorem and its generalisations). For differential operators with finite sequences of polynomial eigenfunctions, we study their restrictions to the space of polynomials of a fixed degree and orthogonality of the eigenfunctions.