Contour Integrations and Parity Results of Hurwitz-type Cyclotomic Euler Sums

TL;DR

This paper explores the parity properties of Hurwitz-type cyclotomic Euler sums using contour integration, deriving explicit formulas and connecting them to cyclotomic multiple Hurwitz polylogarithms, and proposes conjectures on their symmetry.

Contribution

It introduces explicit parity formulas for Hurwitz-type cyclotomic Euler sums and links these sums to cyclotomic multiple Hurwitz polylogarithms, along with conjectures on their symmetry.

Findings

Derived explicit parity formulas for linear, quadratic, and higher-order Euler sums.

Established connections between Euler sums and cyclotomic multiple Hurwitz polylogarithms.

Proposed conjectures on the parity and symmetry of multiple Hurwitz polylogarithm functions.

Abstract

In this paper, we investigate the parity of three class of Hurwitz-type cyclotomic Euler sums using the methods of contour integration and residue computation, and derive explicit parity formulas for linear, quadratic, and some higher-order cases. Based on their connection with cyclotomic multiple Hurwitz polylogarithm functions, we further obtain certain parity results for these functions. At the end of the paper, we propose two conjectures regarding the parity and symmetry of multiple Hurwitz polylogarithm functions of arbitrary depth.