New functional equations of finite multiple polylogarithms

Abstract

We give a finite analogue of the well-known formula ${\mathrm{Li}}_{\underset{n}{\underbrace{1,\dots ,1}}}(t)=\frac{1}{n!}{\mathrm{Li}}_1(t{)}^n$ 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.

9 pages