Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

TL;DR

This paper develops new vector orthogonal polynomial families as eigenfunctions of differential operators, extending Hermite and Laguerre systems, using algebra automorphisms inspired by bispectral operator studies.

Contribution

It introduces novel vector orthogonal polynomials with eigenfunction properties, expanding classical polynomial systems via algebra automorphisms.

Findings

New vector orthogonal polynomial families constructed.

Extension of Hermite and Laguerre polynomial systems.

Identification of algebra automorphisms transforming polynomial systems.

Abstract

We construct new families of vector orthogonal polynomials that have the property to be eigenfunctions of some differential operator. They are extensions of the Hermite and Laguerre polynomial systems. A third family, whose first member has been found by Y. Ben Cheikh and K. Douak is also constructed. The ideas behind our approach lie in the studies of bispectral operators. We exploit automorphisms of associative algebras which transform elementary vector orthogonal polynomial systems which are eigenfunctions of a differential operator into other systems of this type.