Some examples of orthogonal matrix polynomials satisfying odd order differential equations

TL;DR

This paper presents novel examples of matrix orthogonal polynomials that are eigenfunctions of odd-order differential operators, challenging the known restriction to even orders in scalar cases.

Contribution

It introduces the first known weight matrices leading to matrix orthogonal polynomials satisfying odd-order differential equations, expanding the understanding of their spectral properties.

Findings

Examples of weight matrices with odd-order differential eigenoperators.

Matrix orthogonal polynomials not reducible to scalar weights.

New phenomenon in matrix orthogonal polynomial theory.

Abstract

It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form $$ W(t)=t^{\alpha}e^{-t}e^{At}t^{B}t^{B^*}e^{A^* t}, $$ where $A$ and $B$ are certain (nilpotent and diagonal, respectively) $N\times N$ matrices. These weight matrices are the first examples illustrating this new phenomenon which are not reducible to scalar weights.