A Note on Colored Tornheim's Double Series

TL;DR

This paper derives explicit formulas for colored Tornheim's double series using double polylogarithm values at roots of unity, providing new insights into their structure and relations.

Contribution

It introduces a novel explicit formula for colored Tornheim's series, differing from previous results when colors are ±1, and confirms correctness through numerical and algebraic checks.

Findings

Explicit formulas for colored Tornheim's series using polylogarithms

New formulas differ from previous results for certain colors

Numerical and algebraic verification of formulas' correctness

Abstract

In this short note, we provide an explicit formula to compute every colored double Tornheim's series by using double polylogarithm values at roots of unity. When the colors are given by $\pm 1$ our formula is different from that of Tsumura [On alternating analogues of Tornheim's double series II, Ramanujan J. 18 (2009), 81-90] even though numerical data confirm both are correct in almost all the cases. This agreement can also be checked rigorously by using regularized double shuffle relations of the alternating double zeta values in weights less than eight.