Dirichlet type extensions of Euler sums

Ce Xu; Weiping Wang

doi:10.5802/crmath.453

Théorie des nombres

Dirichlet type extensions of Euler sums

Ce Xu ¹ ; Weiping Wang 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

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 979-1010.

Résumé

In this paper, we study the alternating Euler $T$ -sums and $\tilde{S}$ -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 $\tilde{S}$ -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 $\tilde{S}$ -sums are expressible in terms of colored multiple zeta values. Some interesting consequences and illustrative examples are presented.

Reçu le : 2022-07-26
Révisé le : 2022-12-19
Accepté le : 2022-12-19
Publié le : 2023-09-07

DOI : 10.5802/crmath.453

Classification : 11A07, 11M32, 40A25

Affiliations des auteurs :

Ce Xu ¹ ; Weiping Wang 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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     author = {Ce Xu and Weiping Wang},
     title = {Dirichlet type extensions of {Euler} sums},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {979--1010},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.453},
     language = {en},
}

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Ce Xu; Weiping Wang. Dirichlet type extensions of Euler sums. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 979-1010. doi : 10.5802/crmath.453. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.453/

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