Special functions defined by analytic difference equations

We survey our work on a number of special functions that can be viewed as solutions to analytic difference equations. In the infinite-dimensional solution spaces of the pertinent equations, these functions are singled out by various distinctive features. In particular, starting from certain first order difference equations, we consider generalized gamma and zeta functions, as well as Barnes' multiple zeta and gamma functions. Likewise, we review the generalized hypergeometric function we introduced in recent years, emphasizing the four second order Askey-Wilson type difference equations it satisfies. Our results on trigonometric, elliptic and hyperbolic generalizations of the Hurwitz zeta function are presented here for the first time.