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On sum formulas for Mordell-Tornheim zeta values. (English) Zbl 1456.11165

Summary: In this paper we prove new sum formulas for Mordell-Tornheim zeta values in the case of depth 2 and 3, expressing the sums as single multiples of Riemann zeta values. Also, we obtain weighted sum formulas for double Mordell-Tornheim zeta values. Moreover, we present a sum formula for the Mordell-Tornheim series of even arguments.

MSC:

11M32 Multiple Dirichlet series and zeta functions and multizeta values
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References:

[1] J. M. Borwein \andword R. Girgensohn, “Evaluation of triple Euler sums”, Electron. J. Combin. 3:1 (1996), Research Paper 23. \xoxMR1401442 · Zbl 0884.40005
[2] D. M. Bradley \andword X. Zhou, “On Mordell-Tornheim sums and multiple zeta values”, Ann. Sci. Math. Québec 34:1 (2010), 15-23. \xoxMR2744193 \xoxZBL1264.11079 · Zbl 1264.11079
[3] L. Euler, “Meditationes circa singulare serierum genus”, Novi Comm. Acad. Sci. Petropol. 20 (1776), 140-186.
[4] A. Granville, “A decomposition of Riemann”s zeta-function”, \ppword 95-101 \inword Analytic number theory (Kyoto, 1996), \editedbyword Y. Motohashi, London Math. Soc. Lecture Note Ser. 247, Cambridge Univ. Press, 1997. \xoxMR1694987 \xoxZBL0907.11024
[5] L. Guo \andword B. Xie, “Weighted sum formula for multiple zeta values”, J. Number Theory 129:11 (2009), 2747-2765. \xoxMR2549530 \xoxZBL1229.11117 · Zbl 1229.11117
[6] M. E. Hoffman, “Multiple harmonic series”, Pacific J. Math. 152:2 (1992), 275-290. \xoxMR1141796 \xoxZBL0763.11037 · Zbl 0763.11037
[7] J. G. Huard, K. S. Williams\serialcomma \andword N.-Y. Zhang, “On Tornheim”s double series”, Acta Arith. 75:2 (1996), 105-117. \xoxMR1379394 \xoxZBL0858.40008
[8] K. Kamano, “Finite Mordell-Tornheim multiple zeta values”, Funct. Approx. Comment. Math. 54:1 (2016), 65-72. \xoxMR3477735 \xoxZBL1407.11102 · Zbl 1407.11102
[9] T. Machide, “Extended double shuffle relations and the generating function of triple zeta values of any fixed weight”, Kyushu J. Math. 67:2 (2013), 281-307. \xoxMR3115205 \xoxZBL1318.11111 · Zbl 1318.11111
[10] K. Matsumoto, “On Mordell-Tornheim and other multiple zeta-functions”, \inword Proceedings of the Session in Analytic Number Theory and Diophantine Equations, \editedbyword D. R. Heath-Brown \andword B. Z. Moroz, Bonner Math. Schriften 360, Univ. Bonn, 2003. \xoxMR2075634 \xoxZBL1056.11049
[11] Y. Ohno \andword W. Zudilin, “Zeta stars”, Commun. Number Theory Phys. 2:2 (2008), 325-347. \xoxMR2442776 \xoxZBL1228.11132 · Zbl 1228.11132
[12] L. Tornheim, “Harmonic double series”, Amer. J. Math. 72 (1950), 303-314. \xoxMR34860 \xoxZBL0036.17203 · Zbl 0036.17203
[13] H. Tsumura, “On Mordell-Tornheim zeta values”, Proc. Amer. Math. Soc. 133:8 (2005), 2387-2393. \xoxMR2138881 \xoxZBL1059.40003 · Zbl 1059.40003
[14] H. Yuan \andword J. Zhao, “New families of weighted sum formulas for multiple zeta values”, Rocky Mountain J. Math. 45:6 (2015), 2065-2096. \xoxMR3473167 \xoxZBL1332.05010 · Zbl 1332.05010
[15] D. Zagier, “Multiple zeta values”, unpublished manuscript, 1995.
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