Communications in Number Theory and Physics

Volume 15 (2021)

Number 2

On functional equations for Nielsen polylogarithms

Pages: 363 – 454



Steven Charlton (Fachbereich Mathematik (AZ), Universität Hamburg, Germany)

Herbert Gangl (Department of Mathematical Sciences, Durham University, Durham, United Kingdom)

Danylo Radchenko (Department of Mathematics , Eidgenössische Technische Hochschule (ETH) Zurich, Switzerland)


We derive new functional equations for Nielsen polylogarithms. We show that, when viewed $\operatorname{modulo} \mathrm{Li}_5$ 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.


polylogarithms, Nielsen polylogarithms, functional equations, five-term relation, special values

2010 Mathematics Subject Classification

Primary 11G55. Secondary 33E20, 39B32.

This work was initiated during our joint stay at the Kyushu University Multiple Zeta Value Research Center under Masanobu Kaneko’s grant “2017 Kyushu University World Premier International Researchers Invitation Program ‘Progress 100’”. This work continued during the Trimester Program “Periods in Number Theory, Algebraic Geometry and Physics” at the Hausdorff Institute for Mathematics in Bonn. We are grateful to these institutions, as well as to the Max Planck Institute for Mathematics in Bonn, for their hospitality, support and excellent working conditions.

S. Charlton would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program “$K$-theory, algebraic cycles and motivic homotopy theory”, when work on (the final version of) this paper was undertaken.

This work was supported by EPSRC Grant Number EP/R014604/1. S. Charlton and H. Gangl would also like to thank Claude Duhr and Falko Dulat for the $\texttt{PolyLogTools}$ package, which was used for checking various coproduct/coaction calculations, and for helping to lift some identities to numerically checkable ones.

Received 8 November 2019

Accepted 13 January 2021

Published 18 June 2021