Revista de la
Unión Matemática Argentina
A reduction formulafor length-two polylogarithms and some applications
Matilde N. Lalín, Jean-Sébastien Lechasseur
Volume 59, no. 2 (2018), pp. 285–309

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We use shuffle and stuffle relations to give a simple proof of a reduction formula for length-two multiple polylogarithms evaluated in complex parameters of absolute value 1 in terms of a finite sum of products of length-one polylogarithms. This result was originally due to Nakamura and recently reproved by Panzer by different methods. This generalises results of Borwein and Girgensohn for alternating Euler sums and for multiple zeta values twisted by fourth roots of unity by the first author. We also explore implications for other colored multiple zeta values and present some applications to Mahler measure and Feynman diagrams.