Bispectrality for Matrix Laguerre-Sobolev polynomials

TL;DR

This paper investigates bispectral properties of matrix orthogonal polynomials related to Laguerre-Sobolev inner products, revealing new connections via Darboux transformations and Christoffel perturbations.

Contribution

It introduces a novel approach linking Sobolev orthogonal polynomials with matrix orthogonal polynomials through Darboux transformations and Christoffel perturbations.

Findings

Establishes a connection between Sobolev and matrix orthogonal polynomials.

Demonstrates bispectrality of the studied matrix orthogonal polynomials.

Provides an illustrative example with Laguerre-Sobolev polynomials.

Abstract

In this contribution we deal with sequences of polynomials orthogonal with respect to a Sobolev type inner product. A banded symmetric operator is associated with such a sequence of polynomials according to the higher order difference equation they satisfy. Taking into account the Darboux transformation of the corresponding matrix we deduce the connection with a sequence of orthogonal polynomials associated with a Christoffel perturbation of the measure involved in the standard part of the Sobolev inner product. A connection with matrix orthogonal polynomials is stated. The Laguerre-Sobolev type case is studied as an illustrative example. Finally, the bispectrality of such matrix orthogonal polynomials is pointed out.