The Parity of Two Types of Cyclotomic Euler Sums via Contour Integrals

Abstract: In this paper, we employ methods of contour integration and residue calculus to investigate the parity of two classes of cyclotomic Euler-type sums. One class involves products of cyclotomic harmonic numbers, while the other involves products of cyclotomic odd harmonic numbers. We derive explicit formulas for the parity of linear and quadratic cases of these cyclotomic Euler-type sums and provide several illustrative examples. The results for the linear and quadratic cases ensure that we can provide explicit formulas for the parity of cyclotomic multiple $T$-values and cyclotomic multiple $S$-values up to depth three, both of which are even-odd variants of cyclotomic multiple zeta values. Furthermore, we present declarative theorems concerning the parity of these two types of cyclotomic Euler-type sums to arbitrary orders. Additionally, using contour integration techniques, we explore explicit linear and quadratic formulas for these cyclotomic Euler-type sums under more general conditions.