Shen, Zhongyan; Cai, Tianxin Some identities for multiple zeta values. (English) Zbl 1229.11119 J. Number Theory 132, No. 2, 314-323 (2012). Summary: In this note, we obtain the following identities, \[ \sum_{a+b+c=n} \zeta(2a,2b,2c)=\frac 58 \zeta(2n)-\frac 14 \zeta(2)\zeta(2n-2),\quad\text{for}\, n>2, \]\[ \sum_{a+b+c+d=n} \zeta(2a,2b,2c,2d)=\frac{35}{64} \zeta(2n)-\frac 5{16} \zeta(2)\zeta(2n-2),\quad\text{for}\, n>3, \]Meanwhile, some weighted version of sum formulas are also obtained. Cited in 2 ReviewsCited in 19 Documents MSC: 11M32 Multiple Dirichlet series and zeta functions and multizeta values 11B68 Bernoulli and Euler numbers and polynomials Keywords:multiple zeta values; harmonic shuffle relation; Bernoulli numbers PDFBibTeX XMLCite \textit{Z. Shen} and \textit{T. Cai}, J. Number Theory 132, No. 2, 314--323 (2012; Zbl 1229.11119) Full Text: DOI References: [1] Chen, W. Y.C.; Sun, L. H., Extended Zeilbergerʼs algorithm for identities on Bernoulli and Euler polynomials, J. Number Theory, 129, 2111-2132 (2009) · Zbl 1183.11011 [2] Eie, M., A note on Bernoulli numbers and Shintani generalized Bernoulli polynomials, Trans. Amer. Math. Soc., 248, 1117-1136 (1996) · Zbl 0864.11043 [3] Gangl, H.; Kaneko, M.; Zagier, D., Double zeta values and modular forms, (Siegfried, Automorphic Forms and Zeta Functions. In Memory of Tsuneo Arakawa, Proc. of the Conf.. Automorphic Forms and Zeta Functions. In Memory of Tsuneo Arakawa, Proc. of the Conf., Rikkyo University, Tokyo, Japan, 4-7 September 2004 (2006), World Scientific Böherer: World Scientific Böherer Hackensack, NJ), 71-106 · Zbl 1122.11057 [4] Granville, A., A decomposition of Riemannʼs zeta function, (Analytic Number Theory. Analytic Number Theory, London Math. Soc. Lecture Note Ser., vol. 247 (1997), Cambridge University Press), 95-101 · Zbl 0907.11024 [5] Guo, L.; Xie, B., Weighted sum formula for multiple zeta values, J. Number Theory, 129, 2747-2765 (2009) · Zbl 1229.11117 [6] Nakamura, T., Restricted and weighted sum formulas for double zeta values of even weight, Šiauliai Math. Semin., 12, 151-155 (2009) · Zbl 1205.11099 [7] Ohno, Y.; Zudilin, W., Zeta stars, Commun. Number Theory Phys., 2, 327-349 (2008) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.