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Some identities for multiple zeta values. (English) Zbl 1229.11119
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.

11M32Multiple Dirichlet series, etc.
11B68Bernoulli and Euler numbers and polynomials
Full Text: DOI
[1] Chen, W. Y. C.; Sun, L. H.: Extended Zeilberger’s algorithm for identities on Bernoulli and Euler polynomials, J. number theory 129, 2111-2132 (2009) · Zbl 1183.11011 · doi:10.1016/j.jnt.2009.01.026
[2] Eie, M.: A note on Bernoulli numbers and shintani generalized Bernoulli polynomials, Trans. amer. Math. soc. 248, 1117-1136 (1996) · Zbl 0864.11043 · doi:10.1090/S0002-9947-96-01479-1
[3] Gangl, H.; Kaneko, M.; Zagier, D.: Double zeta values and modular forms, Automorphic forms and zeta functions. In memory of tsuneo arakawa, proc. Of the conf., 71-106 (2006) · Zbl 1122.11057
[4] Granville, A.: A decomposition of Riemann’s zeta function, London math. Soc. lecture note ser. 247, 95-101 (1997) · 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 · doi:10.1016/j.jnt.2009.04.018
[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 · http://siauliaims.su.lt/pdfai/2009/Nakamura-09.pdf
[7] Ohno, Y.; Zudilin, W.: Zeta stars, Commun. number theory phys. 2, 327-349 (2008)