Second order differential operators having several families of orthogonal matrix polynomials as eigenfunctions

TL;DR

This paper explores new phenomena in orthogonal matrix polynomials, showing how adding a discrete mass to a weight matrix can produce multiple families sharing common second order differential operators, unlike classical scalar cases.

Contribution

It demonstrates the construction of different orthogonal matrix polynomial families sharing second order differential operators through weight modification.

Findings

Multiple families of matrix polynomials can share eigenfunctions of the same differential operator.

Adding a discrete mass to a weight matrix can induce this property.

Classical scalar orthogonal polynomials do not exhibit this phenomenon.

Abstract

The aim of this paper is to bring into the picture a new phenomenon in the theory of orthogonal matrix polynomials satisfying second order differential equations. The last few years have witnessed some examples of a (fixed) family of orthogonal matrix polynomials whose elements are common eigenfunctions of several linearly independent second order differential operators. We show that the dual situation is also possible: there are examples of different families of matrix polynomials, each family orthogonal with respect to a different weight matrix, whose elements are eigenfunctions of a common second order differential operator. These examples are constructed by adding a discrete mass at certain point to a weight matrix: $\widetilde{W}=W+\delta_{t_0}M(t_0)$. Our method consists in showing how to choose the discrete mass $M(t_0)$, the point $t_0$ where the mass lives and the weight matrix $W$ so that the new weight matrix $\widetilde{W}$ inherits some of the symmetric second order differential operators associated with $W$. It is well known that this situation is not possible for the classical scalar families of Hermite, Laguerre and Jacobi.