The CMV bispectrality of the Jacobi polynomials on the unit circle
Classical Analysis and ODEs
2024-12-17 v1
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.
Keywords
Cite
@article{arxiv.2412.11031,
title = {The CMV bispectrality of the Jacobi polynomials on the unit circle},
author = {Luc Vinet and Alexei Zhedanov},
journal= {arXiv preprint arXiv:2412.11031},
year = {2024}
}
Comments
12 pages, 21 references