Askey-Wilson functions and quantum groups

TL;DR

This paper constructs eigenfunctions of the Askey-Wilson $q$-difference operator using quantum group representations, revealing their integral form and connections to hypergeometric series and elliptic functions.

Contribution

It introduces a novel approach to eigenfunctions of the Askey-Wilson operator via quantum groups, linking them to hypergeometric series and elliptic analogues.

Findings

Eigenfunctions expressed as integral formulas.

Connection to very-well-poised ${}_8\phi_7$-series.

Reduction to elliptic cosine kernel at special parameters.

Abstract

Eigenfunctions of the Askey-Wilson second order $q$-difference operator for $0<q<1$ and $|q|=1$ are constructed as formal matrix coefficients of the principal series representation of the quantized universal enveloping algebra $U_q(sl(2,\mathbb{C}))$. The eigenfunctions are in integral form and may be viewed as analogues of Euler's integral representation for Gauss' hypergeometric series. We show that for $0<q<1$ the resulting eigenfunction can be rewritten as a very-well-poised ${}_8\phi_7$-series, and reduces for special parameter values to a natural elliptic analogue of the cosine kernel.