Tohoku Mathematical Journal
2020

March
SECOND SERIES VOL. 72, NO. 1

Tohoku Math. J.
72 (2020), 149-157

Title NEW FUNCTIONAL EQUATIONS OF FINITE MULTIPLE POLYLOGARITHMS

Author Masataka Ono

(Received October 19, 2017, revised October 22, 2018)
Abstract. We give a finite analogue of the well-known formula $\mathrm{Li}_{\underbrace{1, \ldots, 1}_n}(t)=\frac{1}{n!}\mathrm{Li}_1(t)^n$ of multiple polylogarithms for any positive integer $n$ by using the shuffle relation of finite multiple polylogarithms of Ono--Yamamoto type. Unlike the usual case, the terms regarded as error terms appear in this formula. As a corollary, we obtain “$t \leftrightarrow 1-t$” type new functional equations of finite multiple polylogarithms of Ono--Yamamoto type and Sakugawa-Seki type.

Mathematics Subject Classification. Primary 11M32.

Key words and phrases. Finite multiple polylogarithm, functional equation, shuffle relation.

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