Askey-Wilson functions and quantum groups

Eigenfunctions of the Askey-Wilson second order $q$-difference operator for $0<q<1$ and $\mathrm{\mid }q\mathrm{\mid }=1$ 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 ${}_8{\varphi }_7$-series, and reduces for special parameter values to a natural elliptic analogue of the cosine kernel.