On Transformations of A-Hypergeometric Functions

We propose a systematic study of transformations of A-hypergeometric functions. Our approach is to apply changes of variables corresponding to automorphisms of toric rings, to Euler-type integral representations of A-hypergeometric functions. We show that all linear A-hypergeometric transformations arise from symmetries of the corresponding polytope. As an application of the techniques developed here, we show that the Appell function F4 does not admit a certain kind of Euler-type integral representation.