Comptes Rendus
Théorie des nombres
Dirichlet type extensions of Euler sums
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 979-1010.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.453
Classification : 11A07, 11M32, 40A25

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
@article{CRMATH_2023__361_G6_979_0,
     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},
}
TY  - JOUR
AU  - Ce Xu
AU  - Weiping Wang
TI  - Dirichlet type extensions of Euler sums
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 979
EP  - 1010
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.453
LA  - en
ID  - CRMATH_2023__361_G6_979_0
ER  - 
%0 Journal Article
%A Ce Xu
%A Weiping Wang
%T Dirichlet type extensions of Euler sums
%J Comptes Rendus. Mathématique
%D 2023
%P 979-1010
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.453
%G en
%F CRMATH_2023__361_G6_979_0
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/

[1] Kam Cheong Au Mathematica package MultipleZetaValues (2022) (https://www.researchgate.net/publication/357601353)

[2] Bruce C. Berndt Ramanujan’s Notebooks. Part I, Springer, 1985 | DOI

[3] M. Bigotte; Gerard Jacob; Nour E. Oussous; Michel Petitot Lyndon words and shuffle algebras for generating the coloured multiple zeta values relations tables, Theor. Comput. Sci., Volume 273 (2002) no. 1-2, pp. 271-282 | DOI | MR | Zbl

[4] Johannes Blümlein; David J. Broadhurst; Jos A. M. Vermaseren The multiple zeta value data mine, Comput. Phys., Volume 181 (2010) no. 3, pp. 582-625 | DOI | MR | Zbl

[5] Johannes Blümlein; Stefan Kurth Harmonic sums and Mellin transforms up to two loop order, Phys. Rev. D, Volume 60 (1999) no. 1, 014018

[6] David Borwein; Jonathan M. Borwein On an intriguing integral and some series related to ζ(4), Proc. Am. Math. Soc., Volume 123 (1995) no. 4, pp. 1191-1198 | MR | Zbl

[7] David Borwein; Jonathan M. Borwein; Roland Girgensohn Explicit evaluation of Euler sums, Proc. Edinb. Math. Soc., Volume 38 (1995) no. 2, pp. 277-294 | DOI | MR | Zbl

[8] David J. Broadhurst Multiple zeta values and modular forms in quantum field theory, Computer algebra in quantum field theory. Integration, summation and special functions (Texts and Monographs in Symbolic Computation), Springer, 2013, pp. 33-73 | DOI | MR | Zbl

[9] Hongwei Chen Evaluations of some variant Euler sums, J. Integer Seq., Volume 9 (2006) no. 2, 06.2.3, 9 pages | MR | Zbl

[10] Wenchang Chu Hypergeometric series and the Riemann zeta function, Acta Arith., Volume 82 (1997) no. 2, pp. 103-118 | MR | Zbl

[11] Marc-Antoine Coppo; Bernard Candelpergher Inverse binomial series and values of Arakawa-Kaneko zeta functions, J. Number Theory, Volume 150 (2015), pp. 98-119 | DOI | MR | Zbl

[12] Pieter J. de Doelder On some series containing ψ(x)-ψ(y) and (ψ(x)-ψ(y)) 2 for certain values of x and y, J. Comput. Appl. Math., Volume 37 (1991) no. 1-3, pp. 125-141 | DOI | MR

[13] Philippe Flajolet; Bruno Salvy Euler sums and contour integral representations, Exp. Math., Volume 7 (1998) no. 1, pp. 15-35 | DOI | MR | Zbl

[14] Michael E. Hoffman Multiple harmonic series, Pac. J. Math., Volume 152 (1992) no. 2, pp. 275-290 | DOI | MR | Zbl

[15] Michael E. Hoffman An odd variant of multiple zeta values, Commun. Number Theory Phys., Volume 13 (2019) no. 3, pp. 529-567 | DOI | MR | Zbl

[16] Masanobu Kaneko; Hirofumi Tsumura On multiple zeta values of level two, Tsukuba J. Math., Volume 44 (2020) no. 2, pp. 213-234 | MR | Zbl

[17] Masanobu Kaneko; Hirofumi Tsumura Zeta functions connecting multiple zeta values and poly-Bernoulli numbers, Various aspects of multiple zeta functions (Advanced Studies in Pure Mathematics), Volume 84, Mathematical Society of Japan, 2020, pp. 181-204 | DOI | MR | Zbl

[18] Christian Kassel Quantum Groups, Graduate Texts in Mathematics, 155, Springer, 1995 | DOI

[19] Rudrabhatla Sitaramachandrarao A formula of S. Ramanujan, J. Number Theory, Volume 25 (1987) no. 1, pp. 1-19 | DOI | MR

[20] Ce Xu Some evaluation of parametric Euler sums, J. Math. Anal. Appl., Volume 451 (2017) no. 2, pp. 954-975 | MR | Zbl

[21] Ce Xu Explicit evaluations for several variants of Euler sums, Rocky Mt. J. Math., Volume 51 (2021) no. 3, pp. 1089-1106 | MR | Zbl

[22] Ce Xu Extensions of Euler type sums and Ramanujan type sums, Kyushu J. Math., Volume 75 (2021) no. 2, pp. 295-322 | MR | Zbl

[23] Ce Xu; Weiping Wang Explicit formulas of Euler sums via multiple zeta values, J. Symb. Comput., Volume 101 (2020), pp. 109-127 | MR | Zbl

[24] Ce Xu; Weiping Wang Two variants of Euler sums, Monatsh. Math., Volume 199 (2022) no. 2, pp. 431-454 | MR | Zbl

[25] Don Zagier Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992) (Progress in Mathematics), Volume 120, Birkhäuser, 1994, pp. 497-512 | DOI | MR | Zbl

[26] Jianqiang Zhao Multiple polylogarithm values at roots of unity, C. R. Math. Acad. Sci. Paris, Volume 346 (2008) no. 19-20, pp. 1029-1032 | DOI | Numdam | MR | Zbl

[27] De-Yin Zheng Further summation formulae related to generalized harmonic numbers, J. Math. Anal. Appl., Volume 335 (2007) no. 1, pp. 692-706 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique