  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.
