Jordan algebras and orthogonal polynomials

TL;DR

This paper explores how Jordan algebras can interpret orthogonal polynomials, specifically big -1 Jacobi polynomials, linking algebraic structures to their properties and recurrence relations.

Contribution

It introduces a Jordan algebra framework for understanding big -1 Jacobi polynomials and connects their properties to representations of the algebra.

Findings

Big -1 Jacobi polynomials are eigenfunctions of a Dunkl-type operator.

Recurrence relations are derived from algebra representations.

Ladder operators and Hahn property are established for these polynomials.

Abstract

We illustrate how Jordan algebras can provide a framework for the interpretation of certain classes of orthogonal polynomials. The big -1 Jacobi polynomials are eigenfunctions of a first order operator of Dunkl type. We consider an algebra that has this operator (up to constants) as one of its three generators and whose defining relations are given in terms of anticommutators. It is a special case of the Askey-Wilson algebra AW(3). We show how the structure and recurrence relations of the big -1 Jacobi polynomials are obtained from the representations of this algebra. We also present ladder operators for these polynomials and point out that the big -1 Jacobi polynomials satisfy the Hahn property with respect to a generalized Dunkl operator.