A relativistic hypergeometric function

We survey our work on a function generalizing (2) F-1. This function is a joint eigenfunction of four Askey-Wilsontype 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 II. For a dense subset of II, parameter shifts give rise to an explicit evaluation in terms of rational functions of exponentials ('hyperbolic' functions and plane waves). (c) 2004 Elsevier B.V. All rights reserved.