Euler--Mellin integrals and A-hypergeometric functions

Hypergeometric series representations of Feynman integrals by GKZ hypergeometric systems

“…This discussion involves the perspective of polytopes, which allows a clear and short notation. The theorems are mostly direct implications of the work of [3,54,65] and proofs can be found there. In the description of (1.20) this is equivalent to demanding b j Re ν 0 − m T j • Re ν > 0 for 1 ≤ j ≤ k. Furthermore, if the Newton polytope ∆ G is not full-dimensional, the Feynman integral does not converge absolutely for any choice of ν 0 and ν.…”

“…Thus, we have reformulated the Feynman integral as an Euler-Mellin integral, which defines a meromorphic function in ν ∈ C n+1 . To consider the Euclidean region we have restricted 3 the discussion to the right half space z ∈ C N with Re z j > 0. Further, in corollary 1.3 we had a class of Feynman integrals which provides a simple and analytic solution.…”

Hypergeometric Functions for Projective Toric Curves

“…Viewed as a function of the parameter β, the function M Θ f (x, β) can be meromorphically extended to C 2 ; this is the essence of Hadamard's partie finie [Had32], as understood by Riesz [Rie38]. The explicit description of the process under which the meromorphic extension is obtained, as provided by [BFP12], makes use of combinatorial information that is crucial for our description of the behavior of extended Euler-Mellin integrals.…”

“…For the remainder of this article, we thus focus on the special case of A-hypergeometric systems arising from projective monomial curves, where dedicated tools are available. Specifically, three facts come to our aid for such systems: they have only finitely many rank jumping parameters [CDD99]; for generic parameters, their Euler-Mellin integral solutions span the solution space of the system [BFP12]; and for all parameters, the solutions of the system can be expressed as power series without logarithms [SST00,Sai02]. We deeply rely on the combinatorics of this specific situation to refine the results about hypergeometric integrals and series in order to explain how deformations of special solutions along certain subspaces of the parameter space produce rank jumps.…”

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

“…Just as is the case for Pfaff's transformations, integral expressions for hypergeometric functions provide many of the proofs of the classical transformation formulas. Here we use Euler-type integrals [GKZ90,BFP14] to provide transformations of A-hypergeometric functions. To aid our purposes, we introduce in (2.4) a more symmetric version of these integrals, which has not appeared before.…”