Monomial deformations of certain hypersurfaces and two hypergeometric functions

The purpose of this paper is to give an explicit description, in terms of hypergeometric functions over finite fields, of the zeta functions of certain smooth hypersurfaces that generalize the Dwork family. The key point here is that we count the number of rational points employing both the techniques of character sums and the theory of weights, which enables us to enlighten the calculation of the zeta function.