Pfaffian of Lauricella's hypergeometric system $F_A$

TL;DR

This paper derives a Pfaffian system of differential equations for Lauricella's hypergeometric series $F_A$, utilizing twisted cohomology intersection forms to explicitly describe the connection form, advancing understanding of its structure.

Contribution

It introduces a Pfaffian system for Lauricella's $F_A$ hypergeometric series and employs twisted cohomology intersection forms to explicitly compute the connection form.

Findings

Pfaffian system of rank $2^m$ for $F_A$

Explicit connection form via intersection form

Framework applicable to multivariable hypergeometric functions

Abstract

We give a Pfaffian system of differential equations annihilating Lauricella's hypergeometric series $F_A(a,b,c;x)$ of $m$-variables. This system is integrable of rank $2^m$. To express the connection form of this system, we make use of the intersection form of twisted cohomology groups with respect to integrals representing solutions of this system.