On the all-order epsilon-expansion of generalized hypergeometric functions with integer values of parameters

TL;DR

This paper proves that the epsilon-expansion of generalized hypergeometric functions with integer parameters can be expressed using generalized polylogarithms, providing an efficient method for calculating higher-order coefficients relevant to Feynman diagrams.

Contribution

It introduces a new proof that the epsilon-expansion of such hypergeometric functions can be represented with polylogarithms and offers an efficient algorithm for higher-order coefficient calculation.

Findings

Epsilon-expansion expressed in terms of generalized polylogarithms.

Method provides an efficient algorithm for higher-order coefficients.

Connects hypergeometric functions with Feynman diagram calculations.

Abstract

We continue our study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we apply the approach of obtaining iteratated solutions to the differential equations associated with hypergeometric functions to prove the following result (Theorem 1): The epsilon-expansion of a generalized hypergeometric function with integer values of parameters is expressible in terms of generalized polylogarithms with coefficients that are ratios of polynomials. The method used in this proof provides an efficient algorithm for calculatiing of the higher-order coefficients of Laurent expansion.