Krall–Laguerre commutative algebras of ordinary differential operators

Abstract

In 1999, Grunbaum, Haine and Horozov defined a large family of commutative algebras of ordinary differential operators which have orthogonal polynomials as eigenfunctions. These polynomials are mutually orthogonal with respect to a Laguerre-type weight distribution, thus providing solutions to Krall's problem. In the present paper we give a new proof of their result which establishes a conjecture, concerning the explicit characterization of the dual commutative algebra of eigenvalues. In particular, for the Koornwinder's generalization of Laguerre polynomials, our approach yields an explicit set of generators for the whole algebra of differential operators. We also illustrate how more general Sobolev-type orthogonal polynomials fit within this theory.