On Transformations of <i>A</i>-Hypergeometric Functions

Abstract

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 Ahypergeometric transformations arise from symmetries of the corresponding polytope. As an application of the techniques developed here, we show that the Appell function F 4 does not admit a certain kind of Euler-type integral representation.valid for Re(c) > Re(b) > 0 and |x| < 1, and where B denotes the beta function.On the one hand, transformation formulas are abundant in the hypergeometric literature, see, e.g., reference works such as [AAR99, EMOT53, NIST]. Of particular note is the work of Vidūnas, culminating in [Vid09], which classifies algebraic transformations for the Gauss hypergeometric function. On the other hand, the known hypergeometric transformations mostly involve only a few of the most classical families of hypergeometric functions, essentially those named after Gauss, Appell, and Lauricella.

Feynman integrals as A-hypergeometric functions

“…The proof goes along the lines of Ref. [39] specializing to the nonhomogeneous case and the case of a single polynomial. Let us consider first Eq.(13).…”

“…Solutions of these systems of PDEs are called A-hypergeometric functions. They can be represented as Euler-type integrals and hence these integrals define A-hypergeometric functions [35,[37][38][39]. On the other hand, series solutions can be computed by a generalization of the Frobenius method known as the canonical series algorithm due to Saito, Sturmfels, and Takayama [40].…”