Abstract
In this paper, we extend tools developed in [9] to study Euler T-type sums involving odd harmonic numbers and binomial coefficients. In particular, we will prove that two kinds of Euler T-type sums can be expressed in terms of log(2), zeta values, double T-values, (odd) harmonic numbers and double T-sums.
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References
T. Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J., 153 (1999), 189-209.
D. Borwein, J. M. Borwein and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinburgh Math. Soc., 38 (1995), 277-294.
F. Luo and X. Si, A note on Arakawa-Kaneko zeta values and Kaneko-Tsumura \(\eta\)-values, Bull. Malays. Math. Sci. Soc., 46 (2023), Paper No. 21, 9 pp.
M. E. Hoffman, An odd variant of multiple zeta values, Comm. Number Theory Phys., 13 (2019), 529-567.
M. Kaneko and H. Tsumura, Zeta functions connecting multiple zeta values and poly- Bernoulli numbers, Adv. Stud. Pure Math., 84 (2020), 181-204.
M. Kaneko and H. Tsumura, On multiple zeta values of level two, Tsukuba J. Math., 44 (2020), 213-234.
X. Si, C. Xu and M. Zhang, Quadratic and cubic harmonic number sums, J. Math. Anal. Appl., 447 (2017), 419-434.
X. Si, Euler-type sums involving multiple harmonic sums and binomial coefficients, Open Math., 19 (2021), 1612-1619.
W. Wang and C. Xu, Evaluations of sums involving harmonic numbers and binomial coefficients, J. Difference Equ. Appl., 25 (2019), 1007-1023.
C. Xu, Some evaluation of parametric Euler sums, J. Math. Anal. Appl., 451 (2017), 954-975.
C. Xu, Tornheim type series and nonlinear Euler sums, J. Number Theory, 174 (2017), 40-67.
C. Xu, Identities for the shifted harmonic numbers and binomial coefficients, Filomat, 31 (2017), 6071-6086.
C. Xu, Explicit formulas of some mixed Euler sums via alternating multiple zeta values, Bull. Malays. Math. Sci. Soc., 43 (2020), 3809-3827.
C. Xu, Explicit evaluations for several variants of Euler sums, Rocky Mountain J. Math., 51 (2021), 1089-1106.
C. Xu, Duality formulas for Arakawa-Kaneko zeta values and related variants, Bull. Malays. Math. Sci. Soc., 44 (2021), 3001-3018.
C. Xu and J. Zhao, Variants of multiple zeta values with even and odd summation indices, Math. Z., 300 (2022), 3109-3142.
C. Xu, M. Zhang and W. Zhu, Some evaluation of harmonic number sums, Integral Transforms Spec. Funct., 27 (2016), 937-955.
J. Zhao, Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values, Ser. Number Theory Appl., vol. 12, World Scientific Publishing (Hackensack, NJ, 2016).
Acknowledgement
The authors thank the anonymous referees for suggestions which led to improvements in the exposition.
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Zheng, W., Yang, Y. Evaluations of sums involving odd harmonic numbers and binomial coefficients. Anal Math 50, 323–334 (2024). https://doi.org/10.1007/s10476-024-00011-2
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DOI: https://doi.org/10.1007/s10476-024-00011-2