Elliptic Multiple Polylogarithms with Arbitrary Arguments in \textsc{GiNaC}

TL;DR

This paper introduces a novel algorithm and a GiNaC implementation for the high-precision numerical evaluation of elliptic multiple polylogarithms with arbitrary arguments, leveraging convergent q-series representations.

Contribution

It provides the first public package capable of evaluating elliptic multiple polylogarithms to high precision for arbitrary arguments using a new convergent q-series approach.

Findings

Efficient evaluation of elliptic multiple polylogarithms achieved.

The algorithm converges rapidly in mapped regions.

First public implementation available in GiNaC.

Abstract

We present an algorithm for the numerical evaluation of elliptic multiple polylogarithms for arbitrary arguments and to arbitrary precision. The cornerstone of our approach is a procedure to obtain a convergent $q$-series representation of elliptic multiple polylogarithms. Its coefficients are expressed in terms of ordinary multiple polylogarithms, which can be evaluated efficiently using existing libraries. In a series of preparation steps the elliptic polylogarithms are mapped into a region where the $q$-series converges rapidly. We also present an implementation of our algorithm into the \texttt{GiNaC} framework. This release constitutes the first public package capable of evaluating elliptic multiple polylogarithms to high precision and for arbitrary values of the arguments.