Symmetry Results for Cyclotomic Multiple Hurwitz Zeta Values via Contour Integrals

TL;DR

This paper systematically derives explicit symmetry formulas for cyclotomic multiple Hurwitz zeta values using contour integrals, extending previous symmetry results and providing tools for proving symmetry conjectures.

Contribution

It introduces a direct analytic residue method to obtain symmetry formulas for cyclotomic multiple Hurwitz zeta values without algebraic regularization.

Findings

Derived explicit symmetry formulas for cyclotomic multiple Hurwitz zeta values.

Extended symmetry results to cyclotomic multiple zeta and t-values.

Provided regularization-free formulas applicable in the convergent setting.

Abstract

This paper provides a systematic study of symmetry properties for cyclotomic multiple Hurwitz zeta values with multiple variables and parameters by applying the methods of contour integration and the residue theorem. The main contributions are the derivation of explicit symmetry formulas for cyclotomic multiple (Hurwitz) zeta values, which are obtained directly through analytic residue calculations, without reliance on algebraic regularization. As a concrete application, we deduce analogous symmetry theorems for cyclotomic multiple zeta values and cyclotomic multiple $t$-values. The results extend and complement recent symmetry investigations by Charlton and Hoffman, offering completely explicit and regularization-free formulas in the convergent setting. Moreover, the results of this paper can be used to prove the symmetry conjecture for cyclotomic multiple Hurwitz zeta values with multiple variables and a single parameter. Furthermore, several illustrative corollaries and examples are included, and an open problem concerning possible extensions to other variants of multiple zeta values is posed at the conclusion.