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

TL;DR

The paper proves that the all-order Laurent (ε) expansions of Gauss hypergeometric functions with integer and half-integer parameter shifts, specifically ${}_2F_{1}(I_1+aε,I_2+bε;I_3+cε;z)$ and related variants, can be expressed via Remiddi–Vermaaseren harmonic polylogarithms with rational coefficients. It develops a reduction to a small basis of hypergeometric functions and an efficient iterative algorithm to compute expansion coefficients for both integer and half-integer cases, introducing a conformal transformation and a two-function basis in the half-integer scenario. It also analyzes zero-value upper-parameter cases, provides connections to Nielsen polylogarithms in particular parametrizations, and discusses generalized log-sine functions and colored polylogarithms as part of the expansion structure. The results offer a practical framework for analytic evaluation of ε-expansions in Feynman diagram calculations and clarify the transcendental structure of these hypergeometric functions across different parameter regimes.

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.