Symmetry Results for Cyclotomic Multiple Hurwitz Zeta Values via Contour Integrals

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.

💡 Research Summary

The paper investigates symmetry properties of cyclotomic multiple Hurwitz zeta values (CMHZVs) using contour integration and the residue theorem, avoiding any algebraic regularization. The authors begin by recalling the symmetry conjecture of Charlton and Hoffman for regularized cyclotomic multiple t‑values and set the goal of proving analogous results for the convergent, non‑regularized case of CMHZVs with several variables and parameters.

After establishing notation for ordered sequences, weight, depth, and the definition of CMHZVs \