On Bochner-Krall orthogonal polynomial systems

Abstract

In this paper we address the classical question going back to S. Bochner and H. L. Krall to describe all systems $\{p_{n}(x)\}_{n=0}^\infty$ of orthogonal polynomials (OPS) which are the eigenfunctions of some finite order differential operator. Such systems of orthogonal polynomials are called Bochner-Krall OPS (or BKS for short) and their spectral differential operators are accordingly called Bochner-Krall operators (or BK-operators for short). We show that the leading coefficient of a Nevai type BK-operator is of the form $((x - a)(x-b))^{N/2}$. This settles the special case of the general conjecture 7.3. of [4] describing the leading terms of all BK-operators.