All order epsilon-expansion of Gauss hypergeometric functions with integer and half/integer values of parameters

TL;DR

This paper proves that the Laurent expansion of certain Gauss hypergeometric functions with parameters involving small epsilon can be expressed using harmonic polylogarithms, and provides an efficient algorithm for calculating higher-order coefficients.

Contribution

It introduces a method to express the epsilon-expansion of specific hypergeometric functions in terms of harmonic polylogarithms and develops an efficient algorithm for their higher-order coefficients.

Findings

Laurent expansions are expressible via harmonic polylogarithms.

An efficient algorithm for higher-order coefficients is constructed.

Applicable to hypergeometric functions with integer and half-integer parameters.

Abstract

It is proved that the Laurent expansion of the following Gauss hypergeometric functions, 2F1(I1+a*epsilon, I2+b*ep; I3+c*epsilon;z), 2F1(I1+a*epsilon, I2+b*epsilon;I3+1/2+c*epsilon;z), 2F1(I1+1/2+a*epsilon, I2+b*epsilon; I3+c*epsilon;z), 2F1(I1+1/2+a*epsilon, I2+b*epsilon; I3+1/2+c*epsilon;z), 2F1(I1+1/2+a*epsilon,I2+1/2+b*epsilon; I3+1/2+c*epsilon;z), where I1,I2,I3 are an arbitrary integer nonnegative numbers, a,b,c are an arbitrary numbers and epsilon is an arbitrary small parameters, are expressible in terms of the harmonic polylogarithms of Remiddi and Vermaseren with polynomial coefficients. An efficient algorithm for the calculation of the higher-order coefficients of Laurent expansion is constructed. Some particular cases of Gauss hypergeometric functions are also discussed.