# New functional equations of finite multiple polylogarithms

## TL;DR

This paper establishes new finite analogues of classical multiple polylogarithm formulas, revealing novel functional equations and error term phenomena specific to finite multiple polylogarithms of Ono-Yamamoto type.

## Contribution

It introduces finite analogues of known polylogarithm formulas using shuffle relations, and derives new functional equations unique to the finite setting.

## Key findings

- Finite analogue of the formula for multiple polylogarithms with error terms.
- New functional equations of the form 't ↔ 1 - t' for finite multiple polylogarithms.
- Identification of error terms in finite polylogarithm formulas.

## 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.