Euler and Laplace integral representations of GKZ hypergeometric functions I

Abstract

We introduce an interpolation between Euler integral and Laplace integral: Euler-Laplace integral. We claim that, when parameters $\delta$ of the integrand are non-resonant, the $\mathcal{D}$-module corresponding to Euler-Laplace integral is naturally isomorphic to GKZ hypergeometric system $M_{A}(\delta)$ where $A$ is a generalization of Cayley configuration. As a topological counterpart of this isomorphism, we establish an isomorphism between certain rapid decay homology group and holomorphic solutions of $M_{A}(\delta)$.