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

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.