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

Hongyuan Rui

TL;DR

This work studies the parity properties of three classes of Hurwitz-type cyclotomic Euler sums through contour integration and residue calculus, deriving explicit parity formulas for linear, quadratic, and some higher-order cases. It reveals deep connections between these sums and cyclotomic multiple Hurwitz polylogarithms, yielding parity results for the associated polylogarithm functions and explicit depth-reduction formulas. The results extend the understanding of parity phenomena in cyclotomic Euler-type objects and provide conjectures on parity and symmetry for arbitrary depth, with potential implications for cyclotomic MZVs and their regularization. Overall, the paper advances explicit, residue-based parity structures in Hurwitz-type cyclotomic sums and their polylogarithmic counterparts.

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.

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

TL;DR

This work studies the parity properties of three classes of Hurwitz-type cyclotomic Euler sums through contour integration and residue calculus, deriving explicit parity formulas for linear, quadratic, and some higher-order cases. It reveals deep connections between these sums and cyclotomic multiple Hurwitz polylogarithms, yielding parity results for the associated polylogarithm functions and explicit depth-reduction formulas. The results extend the understanding of parity phenomena in cyclotomic Euler-type objects and provide conjectures on parity and symmetry for arbitrary depth, with potential implications for cyclotomic MZVs and their regularization. Overall, the paper advances explicit, residue-based parity structures in Hurwitz-type cyclotomic sums and their polylogarithmic counterparts.

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.
Paper Structure (7 sections, 17 theorems, 103 equations)

This paper contains 7 sections, 17 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminary Knowledge
  3. Parity Results
  4. Hurwitz-type Cyclotomic Linear Euler Sums
  5. Hurwitz-type Cyclotomic Quadratic Euler Sums
  6. Generalized Hurwitz-type Cyclotomic Euler Sums
  7. Parity Results of Multiple Hurwitz Polylogarithm Function

Key Result

Lemma 2.1

(cf. Flajolet-Salvy) Let $\xi(s)$ be a kernel function and let $r(s)$ be a rational function which is $O(s^{-2})$ at infinity. Then where $S$ is the set of poles of $r(s)$ and $O$ is the set of poles of $\xi(s)$ that are not poles $r(s)$. Here ${\mathop{\rm Re}\nolimits} s{\left( {r(s)},\alpha \right)}$ denotes the residue of $r(s)$ at $s= \alpha.$

Theorems & Definitions (39)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Theorem 3.1
  • ...and 29 more