The GKZ hypergeometric $\mathcal{D}$-module

For an $(n\times N)$-matrix $A$ of rank $n$ with integer entries, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, called the $A$-hypergeometric system. We define the stable GKZ hypergeometric $\mathcal{D}$-module using cohomological functors, which is closely related to the $A$-hypergeometric $\mathcal{D}$-module and the $\mathcal{D}$-module underlying the better behaved GKZ system introduced by Borisov and Horja. We prove the stable GKZ hypergeometric $\mathcal{D}$-module is holonomic and is an integrable connection of rank $n!\mathrm{v}\mathrm{o}\mathrm{l}({\mathrm{\Delta }}_{\mathrm{\infty }})$ on the Zariski open subset parametrizing nondegenerate Laurent polynomials, where ${\mathrm{\Delta }}_{\mathrm{\infty }}$ is the Newton polytope at $\mathrm{\infty }$.