Krall-Laguerre commutative algebras of ordinary differential operators

TL;DR

This paper provides a new proof characterizing the algebra of differential operators with Laguerre-type orthogonal polynomials as eigenfunctions, including explicit generators and extensions to Sobolev-type polynomials.

Contribution

It establishes a conjecture on the dual algebra of eigenvalues and explicitly constructs generators for the algebra related to Koornwinder's Laguerre polynomials.

Findings

Explicit generators for the algebra of differential operators are derived.

The proof confirms the conjecture about the dual algebra of eigenvalues.

Extensions to Sobolev-type orthogonal polynomials are demonstrated.

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.