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)} of orthogonal polynomials (OPS) which are the eigenfunctions of some finite order differential operator, i.e. satisfy the equation \sum_{k=1}^{N}a_{k}(x)y^{(k)}(x)=\la_{n} y(x) (1). 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 say that a BKS has compact type if it is orthogonal with respect to a compactly supported positive measure on the real line. It is well-known that the order N of any BK-operator should be even and every coefficient a_{k}(x) must be a polynomial of degree at most k. Below we show that the leading coefficient of a compact type BK-operator is of the form ((x - a)(x-b))^{N/2}. This settles the special case of the general conjecture of describing the leading terms of all BK-operators. New results on the asymptotic distribution of zeros of polynomial eigenfunctions for a spectral problem (1) are the main ingredient in the proofs.