Evaluating Multiple Polylogarithm Values at Sixth Roots of Unity up to Weight Six

TL;DR

This paper systematically evaluates multiple polylogarithm values at sixth roots of unity up to weight six, providing bases and linear relations for their real and imaginary parts, advancing understanding of these special functions.

Contribution

It introduces explicit bases for the spaces generated by these polylogarithm values and offers a method to express them as linear combinations of basis elements.

Findings

Bases for real and imaginary parts of polylogarithm values are constructed.

A comprehensive table of linear combinations is provided.

Evaluation up to weight six enhances computational and theoretical understanding.

Abstract

We evaluate multiple polylogarithm values at sixth roots of unity up to weight six, i.e. of the form $G(a_1,\ldots,a_w;1)$ where the indices $a_i$ are equal to zero or a sixth root of unity, with $a_1\neq 1$. For $w\leq 6$, we present bases of the linear spaces generated by the real and imaginary parts of $G(a_1,\ldots,a_w;1)$ and present a table for expressing them as linear combinations of the elements of the bases.