Towards all-order Laurent expansion of generalized hypergeometric functions around rational values of parameters

TL;DR

This paper proves that the Laurent expansions of certain generalized hypergeometric functions around rational parameters can be expressed in terms of multiple polylogarithms, providing a systematic way to handle these expansions.

Contribution

It establishes new theorems showing all-order Laurent expansions of hypergeometric functions near rational parameters can be represented using multiple polylogarithms with polynomial ratios.

Findings

Laurent expansions are expressible in terms of multiple polylogarithms.

Expansions involve q-roots of unity and powers of logarithms.

Rational sums are also expressible in terms of multiple polylogarithms.

Abstract

We prove the following theorems: 1) The Laurent expansions in epsilon of the Gauss hypergeometric functions 2F1(I_1+a*epsilon, I_2+b*epsilon; I_3+p/q + c epsilon; z), 2F1(I_1+p/q+a*epsilon, I_2+p/q+b*epsilon; I_3+ p/q+c*epsilon;z), 2F1(I_1+p/q+a*epsilon, I_2+b*epsilon; I_3+p/q+c*epsilon;z), where I_1,I_2,I_3,p,q are arbitrary integers, a,b,c are arbitrary numbers and epsilon is an infinitesimal parameter, are expressible in terms of multiple polylogarithms of q-roots of unity with coefficients that are ratios of polynomials; 2) The Laurent expansion of the Gauss hypergeometric function 2F1(I_1+p/q+a*epsilon, I_2+b*epsilon; I_3+c*epsilon;z) is expressible in terms of multiple polylogarithms of q-roots of unity times powers of logarithm with coefficients that are ratios of polynomials; 3) The multiple inverse rational sums (see Eq. (2)) and the multiple rational sums (see Eq. (3)) are expressible in terms of multiple polylogarithms; 4) The generalized hypergeometric functions (see Eq. (4)) are expressible in terms of multiple polylogarithms with coefficients that are ratios of polynomials.