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

TL;DR

This paper reviews Aomoto's generalized hypergeometric functions on Grassmannian spaces, clarifies their integral representations, and introduces a new systematic approach to Gauss' hypergeometric differential equation.

Contribution

It provides a systematic description of Gauss' hypergeometric differential equation within Aomoto's framework, connecting hypergeometric functions to Grassmannian geometry.

Findings

Integral representations via twisted homology and cohomology clarified.

Detailed analysis of the Gr(2, 4) case.

New systematic first-order Fuchsian differential equation for Gauss' hypergeometric function.

Abstract

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.