A relativistic hypergeometric function (Q1772356)

This paper is a survey of the author's work on a function generalizing the function \({}_2F_1\). 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 \({}_2F_1\). 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 \(\Pi\). For a dense subset of \(\Pi\), parameter shifts give rise to an explicit evaluation in terms of rational functions of exponentials.