  Building on the developments of many people including Evans, Greene, Katz,
McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions
for hypergeometric type algebraic varieties over finite fields and consequently
study hypergeometric functions over finite fields in a manner that is parallel
to that of the classical hypergeometric functions. Using a comparison between
the classical gamma function and its finite field analogue the Gauss sum, we
give a systematic way to obtain certain types of hypergeometric transformation
and evaluation formulas over finite fields and interpret them geometrically
using a Galois representation perspective. As an application, we obtain a few
finite field analogues of algebraic hypergeometric identities, quadratic and
higher transformation formulas, and evaluation formulas. We further apply these
finite field formulas to compute the number of rational points of certain
hypergeometric varieties.
