Matrix differential equations and scalar polynomials satisfying higher order recursions

TL;DR

This paper explores the relationship between scalar differential operators with polynomial eigenfunctions and their matrix-valued counterparts, introducing a new construction method and analyzing the algebraic structures involved.

Contribution

It establishes a canonical method to derive matrix differential operators from scalar ones with polynomial eigenfunctions and investigates the algebraic properties of these matrix operators.

Findings

Constructed matrix differential operators from scalar polynomial eigenfunctions.

Identified new algebraic phenomena in matrix differential operators.

Provided examples illustrating the theoretical framework.

Abstract

We show that any scalar differential operator with a family of polyno- mials as its common eigenfunctions leads canonically to a matrix differen- tial operator with the same property. The construction of the correspond- ing family of matrix valued polynomials has been studied in [D1, D2, DV] but the existence of a differential operator having them as common eigen- functions had not been considered This correspondence goes only one way and most matrix valued situations do not arise in this fashion. We illustrate this general construction with a few examples. In the case of some families of scalar valued polynomials introduced in [GH] we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case.