Non-Symmetric Askey--Wilson Shift Operators

TL;DR

This paper classifies and constructs shift operators for non-symmetric Askey-Wilson polynomials, extending the symmetric case, and explores their applications to norm computation and specializations, linking to non-symmetric Heckman-Opdam polynomials.

Contribution

It introduces new difference-reflection shift operators for non-symmetric Askey-Wilson polynomials and analyzes their properties and specializations.

Findings

Constructed shift operators for non-symmetric Askey-Wilson polynomials.

Computed norms of non-symmetric Askey-Wilson polynomials.

Linked shift operators to non-symmetric Heckman-Opdam polynomials.

Abstract

We classify the shift operators for the symmetric Askey-Wilson polynomials and construct shift operators for the non-symmetric Askey-Wilson polynomials using two decompositions of non-symmetric Askey-Wilson polynomials in terms of symmetric ones. These shift operators are difference-reflection operators, and we discuss the conditions under which they restrict to shift operators for the symmetric Askey-Wilson polynomials. We use them to compute the norms of the non-symmetric Askey-Wilson polynomials and compute their specialisations for $q\to1$. These turn out to be shift operators for the non-symmetric Heckman-Opdam polynomials of type $BC_1$ that have recently been found.