We present a new solution method for a class of first order analytic difference equations. The method yields explicit "minimal" solutions that are essentially unique. Special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic, trigonometric and elliptic type-Euler's gamma function being of rational type. We study these generalized gamma functions in considerable detail. The scattering and weight functions (u· and w-functions)

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... iated to various integrable quantum systems can be expressed in terms of our generalized gamma functions. We obtain detailed information on these u-and w-functions, exploiting the difference equations they satisfy.