Automorphisms of algebras and Bochner`s property for discrete vector orthogonal polynomials

TL;DR

This paper introduces new families of discrete vector orthogonal polynomials as eigenfunctions of difference operators, extending classical systems like Charlier, Meixner, and Kravchuk through algebra automorphisms.

Contribution

It presents novel extensions of Meixner polynomials and a unified algebraic approach to generate such orthogonal polynomial systems.

Findings

New discrete vector orthogonal polynomials with eigenfunction property

Automorphisms of associative algebras transform polynomial systems

Extension of Meixner polynomials is a new result

Abstract

We construct new families of discrete vector orthogonal polynomials that have the property to be eigenfunctions of some difference operator. They are extensions of Charlier, Meixner and Kravchuk polynomial systems. 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 difference operator into other systems of this type. While the extension of Charlier polynomilas is well known it is obtained by different methods. The extension of Meixner polynomial system is new.