Eigenvalue equations for sieved polynomials or proving Askey right again
Classical Analysis and ODEs
2025-07-08 v1
Abstract
The sieved Jacobi polynomials have been introduced by Askey. These can be obtained from conveniently taking to be a root of unity in the Askey-Wilson polynomials. The question of determining if they are eigenfunctions of some operator has been lingering for a long time. Askey impressed on us his conviction that it had an affirmative answer. It is shown that he was right and that this operator is of Dunkl type with cyclic reflections corresponding to the powers of .
Cite
@article{arxiv.2507.03862,
title = {Eigenvalue equations for sieved polynomials or proving Askey right again},
author = {Luc Vinet and Alexei Zhedanov},
journal= {arXiv preprint arXiv:2507.03862},
year = {2025}
}
Comments
14 pages, 1 figure. Contribution to the Richard Askey volume edited by George E. Andrews, Howard S. Cohl, Mourad E. H. Ismail and Luc Vinet. arXiv admin note: text overlap with arXiv:2501.12806