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

2025/09/08 by Zhenlu Wang, Wang, Zhenlu, Ce Xu +1 · 1 citation
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2509.06706

Abstract

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.

Citations

• Contour Integrations and Parity Results of Cyclotomic Euler Sums and Multiple Polylogarithm Function
• An explicit parity theorem for multiple polylogarithms
• An explicit parity theorem for multiple zeta values via multitangent functions
• Weighted and Restricted Sum Formulas of Euler Sums
• Generating functions of multiple t-star values of general level
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• On motivic multiple t values, Saha's basis conjecture, and generators of alternating MZV's
• Dirichlet type extensions of Euler sums
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• Quasi-shuffle products
• Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k

Cited by

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

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