A relativistic hypergeometric function

We survey our work on a function generalizing 2 F 1 . This function is a joint eigenfunction of four Askey–Wilson-type hyperbolic difference operators, reducing to the Askey–Wilson polynomials for certain discrete values of the variables. It is defined by a contour integral generalizing the Barnes representation of 2 F 1 . It has various symmetries, including a hidden D 4 symmetry in the parameters. By means of the associated Hilbert space transform, the difference operators can be promoted to self-adjoint operators, provided the parameters vary over a certain polytope in the parameter space Π . For a dense subset of Π , parameter shifts give rise to an explicit evaluation in terms of rational functions of exponentials (`hyperbolicʹ functions and plane waves).