Some examples of orthogonal matrix polynomials satisfying odd order differential equations

TL;DR

The paper demonstrates that orthogonal matrix polynomials can be eigenfunctions of odd-order differential operators, a phenomenon absent in the scalar case. It introduces nonscalar weight matrices $W(t)=t^{\alpha}e^{-t}e^{At}t^{\frac{1}{2}J}t^{\frac{1}{2}J^*}e^{A^*t}$ with a nilpotent $A$ and diagonal $J$, and develops a framework for symmetry of higher-order operators, including explicit second-order operators and their commutation relations. It then analyzes the algebra of differential operators associated with these weights, showing that for $N=2$ there exist two independent second-order and two independent third-order operators and outlining a generating set and relations for the algebra, while higher $N$ exhibit odd-order operators of orders $2N-1$ and a richer nonscalar structure. These results reveal a broader and more intricate spectral theory for matrix orthogonal polynomials than in the scalar setting, with potential implications for matrix-valued special functions and related algebras.

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 $$ W(t)=t^αe^{-t}e^{At}t^{B}t^{B^*}e^{A^* t}, $$ 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.