Geodesic
Théorie des nombres
Dirichlet type extensions of Euler sums
Xu, Ce1 ; Wang, Weiping2 
1 School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, P.R. China
2 School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, P.R. China
Comptes Rendus. Mathématique, Tome 361 (2023) no. G6, pp. 979-1010

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In this paper, we study the alternating Euler T-sums and S ˜-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler T-sums and S ˜-sums, from which, the parity theorems of Hoffman’s double and triple t-values and Kaneko–Tsumura’s double and triple T-values are further obtained. As supplements, we also show that the linear T-sums and S ˜-sums are expressible in terms of colored multiple zeta values. Some interesting consequences and illustrative examples are presented.

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DOI : 10.5802/crmath.453
Classification : 11A07, 11M32, 40A25

Xu, Ce 1 ; Wang, Weiping 2

1 School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, P.R. China
2 School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, P.R. China
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Dirichlet type extensions of {Euler} sums},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {979--1010},
     publisher = {Acad\'emie des sciences, Paris},
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     number = {G6},
     year = {2023},
     doi = {10.5802/crmath.453},
     language = {en},
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Xu, Ce; Wang, Weiping. Dirichlet type extensions of Euler sums. Comptes Rendus. Mathématique, Tome 361 (2023) no. G6, pp. 979-1010. doi: 10.5802/crmath.453

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