Matrix differential equations and scalar polynomials satisfying higher order recursions

Details Matrix differential equations and scalar polynomials satisfying higher order recursions Matrix differential equations and scalar polynomials satisfying higher order recursions

2008-12-27

arxiv.org/abs/0812.4770v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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.

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