Hypergeometric functions with integer homogeneities

We survey several results on A-hypergeometric systems of linear partial differential equations introduced by Gelfand, Kapranov and Zelevinsky in the case of integer (and thus resonant) parameters, in particular, those differential systems related to sparse systems of polynomial equations. We also study in particular the case of A-hypergeometric systems for which kerA has rank 1. This allows us to clarify the combinatorial meaning of the parameters in one variable classical generalized hypergeometric functions pFp−1, and to describe all such rational functions. 1. Hypergeometric functions Given three complex parameters α, β, γ such that γ / ∈ Z≤0 (or if γ ∈ Z≤0, then α−γ ∈ Z≥1), Gauss hypergeometric function F (α, β, γ;x) was introduced by Gauss in 1812 ([15]). For any natural number n, let (α)n denote the Pochammer symbol (α)n = α · (α+ 1) . . . (α+ n− 1). Note that (1)n = n!. Then, define