The parity theorem for multiple polylogarithms

TL;DR

This paper extends the parity theorem from multiple zeta values to multiple polylogarithms, providing new functional equations, explicit formulas for low depths, and computational tools for special values at roots of unity.

Contribution

It generalizes the parity theorem to multiple polylogarithms and offers explicit formulas and computational methods for special cases.

Findings

Parity theorem extended to MPL at roots of unity

Explicit formulas for depths 2 and 3

Computer program for functional equations

Abstract

We generalize the well-known parity theorem for multiple zeta values (MZV) to functional equations of multiple polylogarithms (MPL). This reproves the parity theorem for MZV with an additional integrality statement, and also provides parity theorems for special values of MPL at roots of unity (also known as coloured MZV). We give explicit formulas in depths 2 and 3 and provide a computer program to compute the functional equations.