$A$-hypergeometric series associated to a lattice polytope with a unique interior lattice point

We associate to lattice points a_0,a_1,...,a_N in Z^n an A-hypergeometric series Φ(λ) with integer coefficients. If a_0 is the unique interior lattice point of the convex hull of a_1,...,a_N, then for every prime p\neq 2 the ratio Φ(λ)/Φ(λ^p) has a p-adic analytic continuation to a closed unit polydisk minus a neighborhood of a hypersurface.

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