The CMV bispectrality of the Jacobi polynomials on the unit circle

TL;DR

This paper demonstrates that Jacobi polynomials on the unit circle are CMV bispectral, satisfying dual eigenvalue problems, and introduces a new algebraic structure called the circle Jacobi algebra to explain their properties.

Contribution

It provides the first explicit example of CMV bispectral orthogonal polynomials on the unit circle and introduces the circle Jacobi algebra as their symmetry structure.

Findings

Jacobi OPUC are CMV bispectral with dual eigenvalue problems.

Introduction of the circle Jacobi algebra as a symmetry algebra.

All properties of Jacobi OPUC derived from algebra representations.

Abstract

We show that the Jacobi polynomials that are orthogonal on the unit circle (the Jacobi OPUC) are CMV bispectral. This means that the corresponding Laurent polynomials in the CMV basis satisfy two dual ordinary eigenvalue problems: a recurrence relation and a differential equation of Dunkl type. This is presumably the first nontrivial explicit example of CMV bispectral OPUC. We introduce the circle Jacobi algebra which plays the role of hidden symmetry algebra for the Jacobi OPUC. All fundamental properties of the Jacobi OPUC can be derived from representations of this algebra.