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Sums of reciprocals of triple binomial coefficients. (English) Zbl 1140.11306

Summary: We investigate the integral representation of infinite sums involving the reciprocals of triple binomial coefficients. We also recover some wellknown properties of \(\zeta \)(3) and extend the range of results given by other authors.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
33B15 Gamma, beta and polygamma functions
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References:

[1] A. Sofo, “General properties involving reciprocals of binomial coefficients,” Journal of Integer Sequences, vol. 9, no. 4, Article ID 06.4.5, 13 pages, 2006. · Zbl 1108.11021
[2] A. Sofo, Computational Techniques for the Summation of Series, Kluwer Academic/Plenum Publishers, New York, NY, USA, 2003. · Zbl 1059.65002
[3] A. Sofo, “Integral representations of ratios of binomial coefficients,” International Journal of Pure and Applied Mathematics, vol. 31, no. 1, pp. 29-46, 2006. · Zbl 1112.11010
[4] A. Sofo, “Some properties of reciprocals of double binomial coefficients,” accepted. · Zbl 1263.11031
[5] http://mathworld.wolfram.com/RiemannZetaFunction.html.
[6] R. Apéry, “Irrationalitè \zeta (2) and \zeta (3),” Astérisque, vol. 61, pp. 11-13, 1979. · Zbl 0401.10049
[7] F. Beukers, “A note on the irrationality of \zeta (2) and \zeta (3),” Bulletin of the London Mathematical Society, vol. 11, no. 3, pp. 268-272, 1979. · Zbl 0421.10023 · doi:10.1112/blms/11.3.268
[8] J. Guillera and J. Sondow, “Double integrals and infinite products for some classical constants via analytic continuations of Lerch/s transcendent,” to appear in The Ramanujan Journal. · Zbl 1216.11075 · doi:10.1007/s11139-007-9102-0
[9] H. Muzaffar, “Some interesting series arising from the power series expansion of (sin - 1x)q,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 14, pp. 2329-2336, 2005. · Zbl 1159.33301 · doi:10.1155/IJMMS.2005.2329
[10] S. B. Ekhad and D. Zeilberger, “A 21st century proof of Dougall/s hypergeometric sum identity,” Journal of Mathematical Analysis and Applications, vol. 147, no. 2, pp. 610-611, 1990. · Zbl 0714.33002 · doi:10.1016/0022-247X(90)90375-P
[11] L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, UK, 1966. · Zbl 0135.28101
[12] S. R. Finch, Mathematical Constants, vol. 94 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 2003. · Zbl 1054.00001
[13] N. Lord, “Problem corner,” The Mathematical Gazette, vol. 89, no. 514, pp. 115-119, 2005.
[14] Z. Nan-Yue and K. S. Williams, “Values of the Riemann zeta function and integrals involving log(2sinh(\theta /2)) and log(2sin(\theta /2)),” Pacific Journal of Mathematics, vol. 168, no. 2, pp. 271-289, 1995. · Zbl 0828.11041
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