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

Elements of Aomoto's generalized hypergeometric functions and a novel perspective on Gauss' hypergeometric differential equation