  We give a definition of generalized hypergeometric functions over finite
fields using modified Gauss sums, which enables us to find clear analogy with
classical hypergeometric functions over the complex numbers. We study their
fundamental properties and prove summation formulas, transformation formulas
and product formulas. An application to zeta functions of K3-surfaces is given.
In the appendix, we give an elementary proof of the Davenport-Hasse
multiplication formula for Gauss sums.
