Hypergeometric Structures in Feynman Integrals

TL;DR

This paper develops automated methods for solving hypergeometric differential equations in Feynman integrals, introducing new sums and series expansions, and providing algorithms for various special functions used in quantum field theory calculations.

Contribution

It presents novel algorithms and software tools for recognizing and solving hypergeometric differential equations in Feynman integrals, including new sums and series expansions.

Findings

Introduction of Hurwitz harmonic sums and their generalizations.

Development of the HypSeries software for series expansions.

Application to various special functions in quantum field theory.

Abstract

Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {\tt Sigma} in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code {\tt HypSeries} transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code {\tt solvePartialLDE} is designed. Generalized hypergeometric functions, Appell-,~Kamp\'e de F\'eriet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton--type functions are considered. We illustrate the algorithms by examples.