Laplace, Residue, and Euler integral representations of GKZ hypergeometric functions

TL;DR

This paper explores various integral and series representations of GKZ hypergeometric functions, introduces new residue integral forms, and establishes their equivalence through $D$-module reformulations.

Contribution

It introduces residue integral representations for GKZ functions and connects all representations via $D$-module theory, providing a unified framework.

Findings

Constructed integration cycles for different representations

Reformulated integral representations using $D$-modules

Established equivalence among various representations

Abstract

We consider four types of representations of solutions of GKZ system: series representations, Laplace integral representations, Euler integral representations, and Residue integral representations which will be introduced in this paper. In the former half of this paper, we provide a method for constructing integration cycles for Laplace, Residue, and Euler integral representations and relate them to series representations. In the latter half, we reformulate our integral representations in terms of direct images of $D$-modules and show their equivalence.