  There exist exactly 166 4-dimensional reflexive polytopes such that the
corresponding 4-dimensional Gorenstein toric Fano varieties have at worst
terminal singularities in codimension 3 and their anticanonical divisor is
divisible by 2. For every such a polytope, one naturally obtains a family of
Fano hypersurfaces X with at worst conifold singularities. A generic
3-dimensional Fano hypersurface X can be interpreted as a flat conifold
degeneration of some smooth Fano 3-folds Y whose classification up to
deformation was obtained by Iskovskikh, Mori and Mukai. In this case, both Fano
varieties X and Y have the same Picard number r. Using toric mirror symmetry,
we define a r-dimensional generalized hypergeometric power series associated to
the dual reflexive polytope. We show that if r =1 then this series is a
normalized regular solution of a modular D3-equation that appears in the
Golyshev correspondence. We expect that the multidimensional power series can
be used to compute the small quantum cohomology ring of all Fano 3-folds Y with
the Picard number r >1 if Y admit a conifold degeneration X.
