Elementary Examples of Solutions to Bochner's Problem for Matrix Differential Operators

TL;DR

This paper introduces an elementary method to generate new solutions to Bochner's problem for matrix differential operators, expanding the known family of solutions and analyzing their algebraic structures.

Contribution

It presents a simple construction technique for solutions to Bochner's problem and describes a large family of solutions derived from classical ones, including explicit generating functions.

Findings

Constructed new solutions from known solutions.

Identified generating functions for matrix orthogonal polynomials.

Analyzed the structure of the algebra D(w).

Abstract

In this paper, we demonstrate an elementary method for constructing new solutions to Bochner's problem for matrix differential operators from known solutions. We then describe a large family of solutions to Bochner's problem, obtained from classical solutions, which include several examples known from the literature. By virtue of the method of construction, we show how one may explicitly identify a generating function for the associated sequence of monic w-orthogonal matrix polynomials p(x,n) as well as the associated algebra D(w) of all matrix differential operators for which the p(x,n) are eigenfunctions. We also include some general results on the structure of the algebra D(w).