  The Gamma-series of Gel'fand-Kapranov-Zelevinsky are adapted so that they
give solutions for certain resonant systems of GKZ hypergeometric differential
equations. For this some complex parameters in the Gamma-series are replaced by
nilpotent elements from a ring $R_{A,T}$. The adapted Gamma-series is a
function $\Psi$ with values in the finite dimensional vector space
$R_{A,T}\otimes C$. Applications of these results in the context of toric
Mirror Symmetry are described. Building on work of Batyrev we show that the
relative cohomology module of a certain hypersurface in a torus is a GKZ
hypergeometric $D$-module which over an appropriate domain is isomorphic to the
trivial $D$-module $R_{A,T}\otimes O_T$, where $O_T$ is the sheaf of
holomorphic functions on this domain. The isomorphism is explicitly given by
adapted Gamma-series. As a result one finds the periods of a holomorphic
differential form of degree $d$ on a $d$-dimensional Calabi-Yau manifold,
needed for the B-model side input to Mirror Symmetry. Relating our work with
that of Batyrev and Borisov we interpret the ring $\cR_{\sA,\gT}$ as the
cohomology ring of a toric variety and a certain principal ideal in it as a
subring of the Chow ring of a Calabi-Yau complete intersection. This
interpretation takes place on the A-model side of Mirror Symmetry.
