Contour Integrations and Parity Results of Cyclotomic Euler $T$-Sums and Multiple $t$-Values
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908537094930432 |
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| author | Wang, Zhenlu Xu, Ce |
| author_facet | Wang, Zhenlu Xu, Ce |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_06706 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Contour Integrations and Parity Results of Cyclotomic Euler $T$-Sums and Multiple $t$-Values Wang, Zhenlu Xu, Ce Number Theory 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. |
| title | Contour Integrations and Parity Results of Cyclotomic Euler $T$-Sums and Multiple $t$-Values |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.06706 |