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http://dx.doi.org/10.4134/CKMS.2015.30.3.227

π AND OTHER FORMULAE IMPLIED BY HYPERGEOMETRIC SUMMATION THEOREMS  

KIM, YONG SUP (Department of Mathematics Education Wonkwang University)
RATHIE, ARJUN KUMAR (Department of Mathematics School of Mathematical and Physical Sciences Central University of Kerala Riverside Transit Campus)
WANG, XIAOXIA (Department of Mathematics Shanghai University)
Publication Information
Communications of the Korean Mathematical Society / v.30, no.3, 2015 , pp. 227-237 More about this Journal
Abstract
By employing certain extended classical summation theorems, several surprising ${\pi}$ and other formulae are displayed.
Keywords
${\pi}$ formula; Gauss summation theorem; Bailey summation theorem; Watson summation theorem; extension summation theorem;
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1 V. Adamchik and S. Wagon, A simple formula for ${\pi}$, Amer. Math. Monthly 104 (1997), no. 9, 852-855.   DOI   ScienceOn
2 D. H. Bailey and J. M. Borwein, Experimental mathematics: examples, methods and implications, Notices Amer. Math. Soc. 52 (2005), no. 5, 502-514.
3 W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935.
4 W. N. Bailey, P. B. Borwein, and S. Plouffe, On the rapid computation of various polylogarithmic constants, Math. Comp. 66 (1997), no. 218, 903-913.   DOI   ScienceOn
5 J. M. Borwein and P. B. Borwein, ${\pi}$ and the AGM, John Wiley & Sons, Inc., New York, 1987.
6 H. C. Chan, More formulas for ${\pi}$, Amer. Math. Monthly 113 (2006), no. 5, 452-455.   DOI   ScienceOn
7 W. Chu, ${\pi}$-formulae implied by Dougall's summation theorem for $_5F_4$-series, Ramanujan J. 26 (2011), no. 2, 251-255.   DOI
8 B. Gourevitch and J. Guillera, Construction of binomial sums for ${\pi}$ and polylogarithmic constant inspired by BBP formulas, Appl. Math. E-Notes 7 (2007), 237-246.
9 J. Guillera, Hypergeometric identities for 10 extended Ramanujan type series, Ramanujan J. 15 (2008), no. 2, 219-234.   DOI
10 J. Guillera, History of the formulas and algorithms for ${\pi}$, Gems in experimental mathematics, 173-188, Contemp. Math., 517, Amer. Math. Soc., Providence, RI, 2010.
11 Y. S. Kim, M. A. Rakha, and A. K. Rathie, Extensions of certain classical summation theorems for the series $_2F_1,\;_3F_2\;and\;_4F_3$ with applications in Ramanujan's summations, Int. J. Math. Math. Sci. 2010 (2010), 309503, 26 pp.
12 J. L. Lavoie, F. Grondin, and A. K. Rathie, Generalizations of Watson's theorem on the sum of a $_3F_2$, Indian J. Math. 32 (1992), no. 1, 23-32.
13 J. L. Lavoie, F. Grondin, and A. K. Rathie, Generalizations of Whipple's theorem on the sum of a $_3F_2$, J. Comput. Appl. Math. 72 (1996), no. 2, 293-300.   DOI   ScienceOn
14 J. L. Lavoie, F. Grondin, A. K. Rathie, and K. Arora, Generalizations of Dixon's theorem on the sum of a $_3F_2$, Math. Comp. 62 (1994), no. 205, 267-276.   DOI
15 A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Gordon and Breach Science, New York, 1986.
16 M. A. Rakha and A. K. Rathie, Generalizations of classical summation theorems for the series $_2F_1\;and\;_3F_2$, Integral Transforms Spec. Funct. 22 (2011), no. 11, 823-840.   DOI
17 R. Vidunas, A generalization of Kummer identity, Rocky Mountain J. Math. 32 (2002), no. 2, 919-936.   DOI   ScienceOn
18 C. Wei, D. Gong, and J. Li, ${\pi}$-Formulas with free parameters, J. Math. Anal. Appl. 396 (2012), no. 2, 880-887.   DOI   ScienceOn
19 W. E. Weisstein, Pi Formulas, MathWorld-A Wolfram Web Resourse, http://mathworld.wolfram.com/PiForm-ulas.html.
20 D. Zheng, Multisection method and further formulae for ${\pi}$, Indian J. Pure Appl. Math. 139 (2008), no. 2, 137-156.