Contour Integrations and Parity Results of Cyclotomic Euler $T$-Sums and Multiple $t$-Values

We will employ the method of contour integration to investigate the parity results of non-embedded cyclotomic multiple $t$-values, which we refer to as cyclotomic Euler $T$-sums. We can provide explicit parity formulas for the linear and quadratic cases of cyclotomic Euler $T$-sums, as well as state a parity theorem for the general case. We also present illustrative examples and corollaries. From this, some parity results for classical cyclotomic multiple $t$-values can be derived. Furthermore, we present several general formulas for cyclotomic Euler $T$-sums with denominators involving arbitrary rational polynomials through residue computations. By evaluating these polynomials and computing residues, many other formulas analogous to cyclotomic Euler $T$-sums can be derived. In particular, we also obtain certain parity results for the cyclotomic versions of multiple $T$-values as defined by Kaneko and Tsumura. Finally, we propose some conjectures and questions regarding the parity of cyclotomic multiple $t$-values and cyclotomic multiple $T$-values.