DOI QR코드

DOI QR Code

DUALITY OF WEIGHTED SUM FORMULAS OF ALTERNATING MULTIPLE T-VALUES

  • Xu, Ce (School of Mathematics and Statistics Anhui Normal University)
  • 투고 : 2020.11.04
  • 심사 : 2021.06.04
  • 발행 : 2021.09.30

초록

Recently, a new kind of multiple zeta value of level two T(k) (which is called multiple T-value) was introduced and studied by Kaneko and Tsumura. In this paper, we define a kind of alternating version of multiple T-values, and study several duality formulas of weighted sum formulas about alternating multiple T-values by using the methods of iterated integral representations and series representations. Some special values of alternating multiple T-values can also be obtained.

키워드

과제정보

The author expresses his deep gratitude to Professors Masanobu Kaneko and Jianqiang Zhao for their valuable discussions and comments. The author thanks the anonymous referee for suggestions which led to improvements in the exposition. The author is supported by the Scientific Research Foundation for Scholars of Anhui Normal University and the University Natural Science Research Project of Anhui Province (Grant No. KJ2020A0057).

참고문헌

  1. H. Gangl, M. Kaneko, and D. Zagier, Double zeta values and modular forms, in Automorphic forms and zeta functions, 71-106, World Sci. Publ., Hackensack, NJ, 2006. https://doi.org/10.1142/9789812774415_0004
  2. M. Gencev, On restricted sum formulas for multiple zeta values with even arguments, Arch. Math. 107 (2016), 9-22. https://doi.org/10.1007/s00013-016-0912-4
  3. A. Granville, A decomposition of Riemann's zeta-function, in Analytic number theory (Kyoto, 1996), 95-101, London Math. Soc. Lecture Note Ser., 247, Cambridge Univ. Press, Cambridge, 1997. https://doi.org/10.1017/CBO9780511666179.009
  4. 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. https://doi.org/10.1142/S1793042115500530
  5. M. E. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), no. 2, 275-290. http://projecteuclid.org/euclid.pjm/1102636166 https://doi.org/10.2140/pjm.1992.152.275
  6. M. E. Hoffman, On multiple zeta values of even arguments, Int. J. Number Theory 13 (2017), no. 3, 705-716. https://doi.org/10.1142/S179304211750035X
  7. M. E. Hoffman, An odd variant of multiple zeta values, Commun. Number Theory Phys. 13 (2019), no. 3, 529-567. https://doi.org/10.4310/CNTP.2019.v13.n3.a2
  8. Z. Li and C. Qin, Weighted sum formulas of multiple zeta values with even arguments, Math. Z. 291 (2019), no. 3-4, 1337-1356. https://doi.org/10.1007/s00209-018-2165-3
  9. Z. Li and C. Xu, Weighted sum formulas of multiple t-values with even arguments, Forum Math. 32 (2020), no. 4, 965-976. https://doi.org/10.1515/forum-2019-0231
  10. M. Kaneko and H. Tsumura, Zeta functions connecting multiple zeta values and poly-Bernoulli numbers, Adv. Stud. Pure Math. 84 (2020), 181-204.
  11. M. Kaneko and H. Tsumura, On multiple zeta values of level two, Tsukuba J. Math. 44 (2020), no. 2, 213-234. https://doi.org/10.21099/tkbjm/20204402213
  12. E. D. Krupnikov and K. S. Kolbig, Some special cases of the generalized hypergeometric function q+1Fq, J. Comput. Appl. Math. 78 (1997), no. 1, 79-95. https://doi.org/10.1016/S0377-0427(96)00111-2
  13. T. Machide, Extended double shuffle relations and the generating function of triple zeta values of any fixed weight, Kyushu J. Math. 67 (2013), no. 2, 281-307. https://doi.org/10.2206/kyushujm.67.281
  14. T. Nakamura, Restricted and weighted sum formulas for double zeta values of even weight, Siauliai Math. Semin. 4(12) (2009), 151-155.
  15. Y. Sasaki, On generalized poly-Bernoulli numbers and related L-functions, J. Number Theory 132 (2012), no. 1, 156-170. https://doi.org/10.1016/j.jnt.2011.07.007
  16. Z. Shen and T. Cai, Some identities for multiple zeta values, J. Number Theory 132 (2012), no. 2, 314-323. https://doi.org/10.1016/j.jnt.2011.06.011
  17. Z. Shen and L. Jia, Some identities for multiple Hurwitz zeta values, J. Number Theory 179 (2017), 256-267. https://doi.org/10.1016/j.jnt.2017.03.003
  18. Z. X. Wang and D. R. Guo, Special Functions, translated from the Chinese by Guo and X. J. Xia, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. https://doi.org/10.1142/0653
  19. W. Wang and C. Xu, Alternating Euler T-sums and Euler S-sums, arXiv:2004.04556.
  20. C. Xu, Some results on multiple polylogarithm functions and alternating multiple zeta values, J. Number Theory 214 (2020), 177-201. https://doi.org/10.1016/j.jnt.2020.04.012
  21. C. Xu and J. Zhao, Variants of multiple zeta values with even and odd summation indices, arXiv.org/2008.13157.
  22. D. Zagier, Values of zeta functions and their applications, in First European Congress of Mathematics, Vol. II (Paris, 1992), 497-512, Progr. Math., 120, Birkhauser, Basel, 1994.
  23. J. Zhao, Sum formula of multiple Hurwitz-zeta values, Forum Math. 27 (2015), no. 2, 929-936. https://doi.org/10.1515/forum-2012-0144
  24. J. Zhao, Multiple zeta functions, multiple polylogarithms and their special values, Series on Number Theory and its Applications, 12, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. https://doi.org/10.1142/9634