Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials

TL;DR

This paper re-expresses nonsymmetric Askey-Wilson polynomials as vector-valued symmetric Laurent polynomials, enabling new insights into their orthogonality and limits to other polynomial families.

Contribution

It introduces a vector-valued formulation of nonsymmetric Askey-Wilson polynomials, allowing analysis of their orthogonality and limit transitions to other polynomial classes.

Findings

Established positive definiteness of the inner product under certain parameters.

Derived a matrix-valued representation of the Dunkl-Cherednik operator.

Facilitated limit transitions to nonsymmetric little q-Jacobi and Jacobi 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.