$\epsilon$-Expansion of Multivariable Hypergeometric Functions Appearing in Feynman Integral Calculus

TL;DR

This paper introduces a computer-implementable method for performing epsilon-expansions of multivariable hypergeometric functions, crucial for Feynman integral calculations, enabling Taylor and Laurent series expansions with explicit coefficient expressions.

Contribution

The authors develop a novel, systematic approach for epsilon-expansion of multivariable hypergeometric functions, applicable to any number of variables, with coefficients expressed as hypergeometric functions.

Findings

Method allows epsilon-expansion in multiple variables

Coefficients are expressed as hypergeometric functions

Applicable to Feynman integral calculations

Abstract

We present a new methodology, suitable for implementation on computer, to perform the $\epsilon$-expansion of hypergeometric functions with linear $\epsilon$ dependent Pochhammer parameters in any number of variables. Our approach allows one to perform Taylor as well as Laurent series expansion of multivariable hypergeometric functions. Each of the coefficients of $\epsilon$ in the series expansion is expressed as a linear combination of multivariable hypergeometric functions with the same domain of convergence as that of the original hypergeometric function. We present illustrative examples of hypergeometric functions in one, two and three variables which are typical of Feynman integral calculus.