Ruijsenaars’ hypergeometric function and the modular double of Uq(sl2(C))

Simultaneous eigenfunctions of two Askey–Wilson second-order difference operators are constructed as formal matrix coefficients of the principal series representation of the modular double of the quantized universal enveloping algebra Uq(sl2(C)). These eigenfunctions are shown to be equal to Ruijsenaars’ hypergeometric function under a proper parameter correspondence.