On functional equations of finite multiple polylogarithms

TL;DR

This paper introduces finite multiple polylogarithms (FMPs), generalizing finite multiple zeta values and polylogarithms, and establishes their functional equations, leading to new supercongruences and evaluations of special values.

Contribution

It presents the first systematic study of FMPs, deriving their functional equations and applying them to compute special values and supercongruences.

Findings

Derived functional equations for FMPs.

Calculated special values involving generalized Bernoulli numbers.

Established supercongruences for Bernoulli numbers.

Abstract

Recently, several people study finite multiple zeta values (FMZVs) and finite polylogarithms (FPs). In this paper, we introduce finite multiple polylogarithms (FMPs), which are natural generalizations of FMZVs and FPs, and we establish functional equations of FMPs. As applications of these functional equations, we calculate special values of FMPs containing generalizations of congruences obtained by Me\v{s}trovi\'c, Z. W. Sun, Z. W. Sun-L. L. Zhao, and Tauraso-J. Zhao. We show supercongruences for certain generalized Bernoulli numbers and the Bernoulli numbers as an appendix.