The CMV bispectrality of the Jacobi polynomials on the unit circle

2024/12/15 by Luc Vinet, Vinet, Luc, Alexei Zhedanov +1 · 1 citation
Mathematics · Physics and Astronomy · #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2412.11031

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.

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• Eigenvalue equations for sieved polynomials or proving Askey right again

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