Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter March 19, 2020

Weighted sum formulas of multiple t-values with even arguments

Zhonghua Li and Ce Xu
From the journal Forum Mathematicum

Abstract

In this paper, we study the weighted sums of multiple t-values and of multiple t-star values at even arguments. Some general weighted sum formulas are given, where the weight coefficients are given by (symmetric) polynomials of the arguments.

MSC 2010: 11M32; 11B68

Communicated by Jan Bruinier


Funding source: China Scholarship Council

Award Identifier / Grant number: 201806310063

Funding statement: The second author is supported by the China Scholarship Council (No. 201806310063).

Acknowledgements

The authors express their deep gratitude to Professor Masanobu Kaneko for valuable discussions and comments.

References

[1] M. Eie, W.-C. Liaw and Y. L. Ong, Several weighted sum formulas of multiple zeta values, Int. J. Number Theory 13 (2017), no. 9, 2253–2264. 10.1142/S1793042117501238Search in Google Scholar

[2] M. Eie and Y. L. Ong, Sum formulas of multiple zeta values with even arguments and polynomial weights, J. Number Theory 188 (2018), 247–262. 10.1016/j.jnt.2018.01.004Search in Google Scholar

[3] H. Gangl, M. Kaneko and D. Zagier, Double zeta values and modular forms, Automorphic Forms and Zeta Functions, World Scientific, Hackensack (2006), 71–106. 10.1142/9789812774415_0004Search in Google Scholar

[4] M. Genčev, On restricted sum formulas for multiple zeta values with even arguments, Arch. Math. 107 (2016), 9–22. 10.1007/s00013-016-0912-4Search in Google Scholar

[5] L. Guo, P. Lei and J. Zhao, Families of weighted sum formulas for multiple zeta values, Int. J. Number Theory 11 (2015), no. 3, 997–1025. 10.1142/S1793042115500530Search in Google Scholar

[6] M. E. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), no. 2, 275–290. 10.2140/pjm.1992.152.275Search in Google Scholar

[7] M. E. Hoffman, On multiple zeta values of even arguments, Int. J. Number Theory 13 (2017), no. 3, 705–716. 10.1142/S179304211750035XSearch in Google Scholar

[8] M. E. Hoffman, An odd variant of multiple zeta values, Commun. Number Theory Phys. 13 (2019), no. 3, 529–567. 10.4310/CNTP.2019.v13.n3.a2Search in Google Scholar

[9] Y. Komori, K. Matsumoto and H. Tsumura, A study on multiple zeta values from the viewpoint of zeta-functions of root systems, Funct. Approx. Comment. Math. 51 (2014), no. 1, 43–76. 10.7169/facm/2014.51.1.3Search in Google Scholar

[10] Z. Li and C. Qin, Some relations deduced from regularized double shuffle relations of multiple zeta values, preprint (2016), https://arxiv.org/abs/1610.05480. 10.1142/S1793042121500068Search in Google Scholar

[11] Z. Li and C. Qin, Weighted sum formulas of multiple zeta values with even arguments, Math. Z. 291 (2019), no. 3–4, 1337–1356. 10.1007/s00209-018-2165-3Search in Google Scholar

[12] T. Nakamura, Restricted and weighted sum formulas for double zeta values of even weight, Šiauliai Math. Semin. 4(12) (2009), 151–155. Search in Google Scholar

[13] Z. Shen and T. Cai, Some identities for multiple zeta values, J. Number Theory 132 (2012), no. 2, 314–323. 10.1016/j.jnt.2011.06.011Search in Google Scholar

[14] Z. Shen and L. Jia, Some identities for multiple Hurwitz zeta values, J. Number Theory 179 (2017), 256–267. 10.1016/j.jnt.2017.03.003Search in Google Scholar

[15] Z. Y. Shen and T. X. Cai, Some identities for alternating multiple zeta values, Acta Math. Sinica (Chin. Ser.) 56 (2013), no. 4, 441–450. 10.1016/j.jnt.2021.04.005Search in Google Scholar

[16] D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics. Vol. II (Paris 1992), Progr. Math. 120, Birkhäuser, Basel (1994), 497–512. 10.1007/978-3-0348-9112-7_23Search in Google Scholar

[17] J. Zhao, Sum formula of multiple Hurwitz-zeta values, Forum Math. 27 (2015), no. 2, 929–936. 10.1515/forum-2012-0144Search in Google Scholar

[18] J. Zhao, Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values, Ser. Number Theory Appl. 12, World Scientific, Hackensack, 2016. 10.1142/9634Search in Google Scholar

Received: 2019-08-22
Revised: 2020-03-03
Published Online: 2020-03-19
Published in Print: 2020-07-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Scroll Up Arrow