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Integer powers of arcsin. (English) Zbl 1137.33300

Summary: New simple nested-sum representations for powers of the arcsin function are given. This generalization of Ramanujan’s work makes connections to finite binomial sums and polylogarithms.

MSC:

33B10 Exponential and trigonometric functions

Software:

OEIS; lsjk
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References:

[1] J. M. Borwein and D. Bailey, Mathematics by Experiment, A K Peters, Natick, Mass, USA, 2004. · Zbl 1083.00001
[2] J. M. Borwein, D. Bailey, and R. Girgensohn, Experimentation in Mathematics, A K Peters, Natick, Mass, USA, 2004. · Zbl 1083.00002
[3] E. W. Weisstein, “Inverse Sine,” from MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/InverseSine.html.
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[5] J. Edwards, Differential Calculus, MacMillan, London, UK, 2nd edition, 1982.
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[9] M. Yu. Kalmykov and A. Sheplyakov, “lsjk-a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine functions,” Computer Physics Communications, vol. 172, no. 1, pp. 45-59, 2005. · Zbl 05801549 · doi:10.1016/j.cpc.2005.04.013
[10] D. Borwein, J. M. Borwein, and R. E. Crandall, “Effective laguerre asymptotics,” submitted to SIAM Journal on Numerical Analysis. · Zbl 1196.33010
[11] J. M. Borwein, D. J. Broadhurst, and J. Kamnitzer, “Central binomial sums, multiple Clausen values, and zeta values,” Experimental Mathematics, vol. 10, no. 1, pp. 25-34, 2001. · Zbl 0998.11045 · doi:10.1080/10586458.2001.10504426
[12] N. J. Sloane, “Online Encyclopedia of Integer Sequences,” (http://www.research.att.com/ njas/sequences/). · Zbl 1274.11001
[13] L. Lewin, Polylogarithms and Associated Functions, North-Holland, New York, NY, USA, 1981. · Zbl 0465.33001
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