Askey-Wilson Polynomials and Branching Laws

TL;DR

This paper derives new connection coefficient formulas for nonsymmetric Askey-Wilson polynomials when shifting parameters, linking special functions with representation theory and providing a new proof of a classical result.

Contribution

It introduces novel connection coefficient formulas for parameter shifts in nonsymmetric Askey-Wilson polynomials and demonstrates their application in re-proving a known symmetric case result.

Findings

Derived connection coefficient formulas for parameter shifts.

Re-proved a classical Askey-Wilson symmetric case result.

Utilized Hecke algebra actions and co-cycle conditions in proofs.

Abstract

Connection coefficient formulas for special functions describe change of basis matrices under a parameter change, for bases formed by the special functions. Such formulas are related to branching questions in representation theory. The Askey-Wilson polynomials are one of the most general 1-variable special functions. Our main results are connection coefficient formulas for shifting one of the parameters of the nonsymmetric Askey-Wilson polynomials. We also show how one of these results can be used to re-prove an old result of Askey and Wilson in the symmetric case. The method of proof combines establishing a simpler special case of shifting one parameter by a factor of q with using a co-cycle condition property of the transition matrices involved. Supporting computations use the Noumi representation and are based on simple formulas for how some basic Hecke algebra elements act on natural almost symmetric Laurent polynomials.