e-mail: siru@wxs.nl

**A `relativistic' hypergeometric function obeying four
Askey-Wilson type difference equations**

**Abstract**

As is well known, the hypergeometric function ${}_2F_1$ admits a contour integral representation (due to Barnes), which involves Euler's gamma function. We present a novel generalization of ${}_2F_1$ via a Barnes type representation, involving a generalized gamma function. Our ${}_2F_1$-generalization is a simultaneous eigenfunction of four independent hyperbolic difference operators of Askey-Wilson type. In contrast to ${}_2F_1$, it is meromorphic in all of its variables. Moreover, it has various remarkable symmetry properties that are not preserved for its $q \to 1$ (or `nonrelativistic') limit ${}_2F_1$.

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