On functional equations for Nielsen polylogarithms

TL;DR

This paper derives new functional equations for Nielsen polylogarithms, revealing their relations to classical polylogarithm identities and extending known results up to weight 8.

Contribution

It introduces novel functional equations for Nielsen polylogarithms and connects them to classical polylogarithm relations, especially at weight 5 and higher.

Findings

Weight 5 Nielsen polylogarithm satisfies the dilogarithm five-term relation modulo lower weight functions.

Functional equations and evaluations are provided for weights up to 8.

General families of identities are established for higher weights.

Abstract

We derive new functional equations for Nielsen polylogarithms. We show that, when viewed modulo $\mathrm{Li}_5$ and products of lower weight functions, the weight $5$ Nielsen polylogarithm $S_{3,2}$ satisfies the dilogarithm five-term relation. We also give some functional equations and evaluations for Nielsen polylogarithms in weights up to 8, and general families of identities in higher weight.