Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 1

Special Issue in honor of Don Zagier

Guest editors: Benedict H. Gross, Ken Ono, and Fernando Rodriguez Villegas

Functional equations of polygonal type for multiple polylogarithms in weights $5$, $6$ and $7$

Pages: 85 – 93

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n1.a5


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

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

Danylo Radchenko (Laboratoire Paul Painlevé, Université de Lille, Villeneuve d’Ascq, France)


We present new functional equations of multiple polylogarithms in weights $5$, $6$ and $7$ and use them for explicit depth reduction. These identities generalize the crucial identity $\mathbf{Q}_4$ from the recent work of Goncharov and Rudenko that was used in their proof of the weight $4$ case of Zagier’s Polylogarithm Conjecture.


polylogarithms, functional equations, cluster relations, Zagier’s Polylogarithm Conjecture

2010 Mathematics Subject Classification

Primary 11G55. Secondary 33E20, 39B32.

Received 4 November 2021

Accepted 17 January 2022

Published 3 April 2023