New functional equations of finite multiple polylogarithms

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.