• 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)
  • Received : 2015.04.16
  • Published : 2015.06.30


By employing certain extended classical summation theorems, several surprising ${\pi}$ and other formulae are displayed.


${\pi}$ formula;Gauss summation theorem;Bailey summation theorem;Watson summation theorem;extension summation theorem


Supported by : Wonkwang University


  1. V. Adamchik and S. Wagon, A simple formula for ${\pi}$, Amer. Math. Monthly 104 (1997), no. 9, 852-855.
  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.
  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.
  7. W. Chu, ${\pi}$-formulae implied by Dougall's summation theorem for $_5F_4$-series, Ramanujan J. 26 (2011), no. 2, 251-255.
  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.
  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.
  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.
  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.
  17. R. Vidunas, A generalization of Kummer identity, Rocky Mountain J. Math. 32 (2002), no. 2, 919-936.
  18. C. Wei, D. Gong, and J. Li, ${\pi}$-Formulas with free parameters, J. Math. Anal. Appl. 396 (2012), no. 2, 880-887.
  19. W. E. Weisstein, Pi Formulas, MathWorld-A Wolfram Web Resourse,
  20. D. Zheng, Multisection method and further formulae for ${\pi}$, Indian J. Pure Appl. Math. 139 (2008), no. 2, 137-156.