Euler and Laplace integral representations of GKZ hypergeometric functions II

Abstract

We introduce an interpolation between Euler integral and Laplace integral: Euler-Laplace integral. We show, when parameters $d$ of the integrand is non-resonant, the $\mathcal{D}$-module corresponding to Euler-Laplace integral is naturally isomorphic to GKZ hypergeometric system $M_A(d)$ 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 $M_A(d)$. Based on these foundations, we give a combinatorial method of constructing a basis of rapid decay cycles by means of regular triangulations. The remarkable feature of this construction is that this basis of cycles is explicitly related to $\Gamma$-series solutions. In the last part, we concentrate on Euler integral representations. We determine the homology intersection matrix with respect to our basis of cycles when the regular triangulation is unimodular. As an application, we obtain closed formulae of the quadratic relations of Aomoto-Gelfand hypergeometric functions in terms of bipartite graphs.