  We review Aomoto's generalized hypergeometric functions on Grassmannian
spaces Gr(k +1, n+1). Particularly, we clarify integral representations of the
generalized hypergeometric functions in terms of twisted homology and
cohomology. With an example of the Gr(2, 4) case, we consider in detail Gauss'
original hypergeometric functions in Aomoto's framework. This leads us to
present a new systematic description of Gauss' hypergeometric differential
equation in a form of a first order Fuchsian differential equation.
