Duality and difference operators for matrix valued discrete polynomials on the nonnegative integers

TL;DR

This paper introduces a duality concept for matrix-valued orthogonal polynomials on nonnegative integers, linking dual families to difference operators within Fourier algebras, and provides explicit examples with Charlier-type polynomials.

Contribution

It develops a new duality framework for matrix-valued orthogonal polynomials and constructs explicit difference operators and recurrence relations for Charlier-type examples.

Findings

Dual families are related to difference operators in Fourier algebras.

Explicit formulas for three-term recurrences and norms of Charlier-type polynomials.

Construction of dual families using shift operators and difference equations.

Abstract

In this paper we introduce a notion of duality for matrix valued orthogonal polynomials with respect to a measure supported on the nonnegative integers. We show that the dual families are closely related to certain difference operators acting on the matrix orthogonal polynomials. These operators belong to the so called Fourier algebras, which play a key role in the construction of the families. In order to illustrate duality, we describe a family of Charlier type matrix orthogonal polynomials with explicit shift operators which allow us to find explicit formulas for three term recurrences, difference operators and square norms. These are the essential ingredients for the construction of different dual families.