Askey-wilson Functions and Quantum Groups

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 Uq(sl(2,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 8φ7-series, and reduces for special parameter values to a natural elliptic analogue of the cosine kernel. Dedicated to Mizan Rahman