Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials
Abstract
Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is represented as a 2x2 matrix-valued operator. As a new result made possible by this approach we obtain positive definiteness of the inner product in the orthogonality relations, under certain constraints on the parameters. A limit transition to nonsymmetric little q-Jacobi polynomials also becomes possible in this way. Nonsymmetric Jacobi polynomials are considered as limits both of the Askey-Wilson and of the little q-Jacobi case.
Cite
@article{arxiv.1006.1140,
title = {Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials},
author = {Tom H. Koornwinder and Fethi Bouzeffour},
journal= {arXiv preprint arXiv:1006.1140},
year = {2018}
}
Comments
16 pages. Dedicated to Paul Butzer on the occasion of his 80th birthday. v4: minor correction in (4.14)