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We formulate general conjectures about the relationship between the A-modelconnection on the cohomology of a $d$-dimensional Calabi-Yau completeintersection $V$ of $r$ hypersurfaces $V_1, \ldots, V_r$ in a toric variety${\bf P}_{\Sigma}$ and the system of differential operators annihilating thespecial hypergeometric function $\Phi_0$ depending on the fan $\Sigma$. In thiscontext, the Mirror Symmetry phenomenon can be interpreted as the followingtwofold characterization of the series $\Phi_0$. First, $\Phi_0$ is defined byintersection numbers of rational curves in ${\bf P}_{\Sigma}$ with thehypersurfaces $V_i$ and their toric degenerations. Second, $\Phi_0$ is thepower expansion near a boundary point of the moduli space of the monodromyinvariant period of the holomorphic differential $d$-form on an anotherCalabi-Yau $d$-fold $V'$ which is called Mirror of $V$. Using the generalizedhypergeometric series, we propose a general construction for Mirrors $V'$ of$V$ and canonical $q$-coordinates on the moduli spaces of Calabi-Yau manifolds.Comment: 45 pages, Latex 2.0

Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties