Expansion of hypergeometric functions in terms of polylogarithms with nontrivial variable change

TL;DR

This paper develops a method to expand hypergeometric functions with rational indices into polylogarithms using differential equations and variable transformations, supported by a Mathematica package.

Contribution

It introduces a systematic approach for expanding hypergeometric functions with rational indices into Goncharov polylogarithms, including nontrivial variable changes, with an implementation in Mathematica.

Findings

Effective reduction to canonical basis achieved

Expansion formulas for hypergeometric functions derived

Mathematica package Diogenes provided for implementation

Abstract

Hypergeometric functions of one and many variables play an important role in various branches of modern physics and mathematics. Often we have hypergeometric functions with indices linear dependent on a small parameter with respect to which one needs to perform Laurent expansions. Moreover such expansions are desirable to be expressed in terms of well known functions which can be evaluated with arbitrary precision. To solve this problem we use the differential equation method and the reduction of corresponding differential systems to canonical basis. Specifically we will be interested in the generalized hypergeometric functions of one variable together with Appell and Lauricella functions and their expansions in terms of Goncharov polylogarithms. Particular attention will be given to the case of rational indices of considered hypergeometric functions when the reduction to canonical basis involves nontrivial variable change. The article comes with a Mathematica package Diogenes, which provides algorithmic implementation of the required steps.