Mellin–Barnes integrals as Fourier–Mukai transforms

We study the generalized hypergeometric system introduced by Gelfand, Kapranov and Zelevinsky and its relationship with the toric Deligne–Mumford (DM) stacks recently studied by Borisov, Chen and Smith. We construct series solutions with values in a combinatorial version of the Chen–Ruan (orbifold) cohomology and in the K-theory of the associated DM stacks. In the spirit of the homological mirror symmetry conjecture of Kontsevich, we show that the K-theory action of the Fourier–Mukai functors associated to basic toric birational maps of DM stacks are mirrored by analytic continuation transformations of Mellin–Barnes type.