# Contour Integrations and Parity Results of Cyclotomic Euler Sums and Multiple Polylogarithm Function

## TL;DR

This paper uses contour integration to analyze the parity properties of cyclotomic Euler sums and multiple polylogarithms, providing explicit formulas and extending known results in the field.

## Contribution

It introduces a novel approach using contour integrals to establish parity results for various cyclotomic Euler sums and polylogarithms, including new formulas for cubic sums.

## Key findings

- Parity results for cyclotomic Euler sums of arbitrary order
- Explicit formulas for linear and quadratic cyclotomic Euler sums
- Formulas for parity of cyclotomic cubic Euler sums and polylogarithms

## Abstract

In this paper, we define extended trigonometric functions via series and employ the method of contour integration to investigate the parity of certain cyclotomic Euler sums and multiple polylogarithm function. We can provide the statement of parity results for cyclotomic Euler sums of arbitrary order, explicit formulas for the parity of cyclotomic linear and quadratic Euler sums, as well as some formulas for the parity of cyclotomic cubic Euler sums and multiple polylogarithms. As a direct corollary, we derive known formulas concerning the parity of classical Euler sums and alternating Euler sums.