Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams

TL;DR

This paper develops methods to reduce Gauss hypergeometric functions with arbitrary parameters to simpler forms, classifies functions with integer or half-integer parameters, and constructs epsilon-expansions relevant for Feynman diagram calculations.

Contribution

It introduces a reduction technique for hypergeometric functions with arbitrary parameters and provides epsilon-expansions for specific types, including new functions related to elliptic functions.

Findings

Reduction of hypergeometric functions to fixed-parameter forms

Classification into three algebraically independent types for integer/half-integer parameters

Explicit epsilon-expansions up to weight 4 expressible via Nielsen polylogarithms

Abstract

The Gauss hypergeometric functions 2F1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown that in the case of integer and/or half-integer values of parameters there are only three types of algebraically independent Gauss hypergeometric functions. The epsilon-expansion of functions of one of this type (type F in our classification) demands the introduction of new functions related to generalizations of elliptic functions. For the five other types of functions the higher-order epsilon-expansion up to functions of weight 4 are constructed. The result of the expansion is expressible in terms of Nielsen polylogarithms only. The reductions and epsilon-expansion of q-loop off-shell propagator diagrams with one massive line and q massless lines and q-loop bubble with two-massive lines and q-1 massless lines are considered. The code (Mathematica/FORM) is available via the www at this URL http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html