Acknowledgement
Xiaoxue Wei is supported by the Natural Science Foundation (Grant No. Anhui Province 2108085QG304). Ce Xu is supported by the National Natural Science Foundation of China (Grant No. 12101008), the Natural Science Foundation of Anhui Province (Grant No. 2108085QA01) and the University Natural Science Research Project of Anhui Province (Grant No. KJ2020A0057).
References
- H. Alzer and J. Choi, Four parametric linear Euler sums, J. Math. Anal. Appl. 484 (2020), no. 1, 123661, 22 pp. https://doi.org/10.1016/j.jmaa.2019.123661
- D. H. Bailey, J. M. Borwein, and R. Girgensohn, Experimental evaluation of Euler sums, Experiment. Math. 3 (1994), no. 1, 17-30. http://projecteuclid.org/euclid.em/1062621000 1062621000
- D. Borwein, J. M. Borwein, and D. M. Bradley, Parametric Euler sum identities, J. Math. Anal. Appl. 316 (2006), no. 1, 328-338. https://doi.org/10.1016/j.jmaa.2005.04.040
- J. Choi, Certain summation formulas involving harmonic numbers and generalized harmonic numbers, Appl. Math. Comput. 218 (2011), no. 3, 734-740. https://doi.org/10.1016/j.amc.2011.01.062
- J. Choi, Finite summation formulas involving binomial coefficients, harmonic numbers and generalized harmonic numbers, J. Inequal. Appl. 2013, 2013:49, 11 pp. https://doi.org/10.1186/1029-242X-2013-49
- J. Choi and H. M. Srivastava, Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Comput. Modelling 54 (2011), no. 9-10, 2220-2234. https://doi.org/10.1016/j.mcm.2011.05.032
- L. Euler, Meditationes circa singulare serierum genus, Novi Comm. Acad. Sci. Petropol. 20 (1776), 140-186; reprinted in Opera Omnia, Ser. Ib 15(1927), 217-267, B. Teubner (ed.), Berlin.
- P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experiment. Math. 7 (1998), no. 1, 15-35. http://projecteuclid.org/euclid.em/1047674270 1047674270
- M. E. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), no. 2, 275-290. http://projecteuclid.org/euclid.pjm/1102636166 102636166
- 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
- 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
- F. Luo and X. Si, A note on Arakawa-Kaneko zeta values and Kaneko-Tsumura 𝜂-values, Bull. Malays. Math. Sci. Soc. 46 (2023), no. 1, Paper No. 21, 9 pp. https://doi.org/10.1007/s40840-022-01420-y
- I. Mezo, Nonlinear Euler sums, Pacific J. Math. 272 (2014), no. 1, 201-226. https://doi.org/10.2140/pjm.2014.272.201
- X. Si, Euler-type sums involving multiple harmonic sums and binomial coefficients, Open Math. 19 (2021), no. 1, 1612-1619. https://doi.org/10.1515/math-2021-0124
- A. Sofo, Quadratic alternating harmonic number sums, J. Number Theory 154 (2015), 144-159. https://doi.org/10.1016/j.jnt.2015.02.013
- A. Sofo and J. Choi, Extension of the four Euler sums being linear with parameters and series involving the zeta functions, J. Math. Anal. Appl. 515 (2022), no. 1, Paper No. 126370, 23 pp. https://doi.org/10.1016/j.jmaa.2022.126370
- A. Sofo and H. M. Srivastava, Identities for the harmonic numbers and binomial coefficients, Ramanujan J. 25 (2011), no. 1, 93-113. https://doi.org/10.1007/s11139-010-9228-3
- W. Wang and Y. Lyu, Euler sums and Stirling sums, J. Number Theory 185 (2018), 160-193. https://doi.org/10.1016/j.jnt.2017.08.037
- C. Xu, Some evaluation of parametric Euler sums, J. Math. Anal. Appl. 451 (2017), no. 2, 954-975. https://doi.org/10.1016/j.jmaa.2017.02.047
- C. Xu, Some evaluations of infinite series involving parametric harmonic numbers, Int. J. Number Theory 15 (2019), no. 7, 1531-1546. https://doi.org/10.1142/S179304211950088X
- C. Xu and W. Wang, Explicit formulas of Euler sums via multiple zeta values, J. Symbolic Comput. 101 (2020), 109-127. https://doi.org/10.1016/j.jsc.2019.06.009
- C. Xu and J. Zhao, Variants of multiple zeta values with even and odd summation indices, Math. Z. 300 (2022), no. 3, 3109-3142. https://doi.org/10.1007/s00209-021-02889-2
- H. Yuan and J. Zhao, Double shuffle relations of double zeta values and the double Eisenstein series at level N, J. Lond. Math. Soc. (2) 92 (2015), no. 3, 520-546. https://doi.org/10.1112/jlms/jdv042
- D. Zagier, Values of Zeta Functions and Their Applications, First European Congress of Mathematics, Vol. II (Paris, 1992), 497-512, Progr. Math., 120, Birkhauser, Basel, 1994.
- J. Zhao, Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values, Series on Number Theory and its Applications, 12, World Sci. Publ., Hackensack, NJ, 2016. https://doi.org/10.1142/9634