Advertisement

The Ramanujan Journal

, Volume 26, Issue 2, pp 251–255 | Cite as

π-formulas implied by Dougall’s summation theorem for 5F4-series

  • Wenchang Chu
Article

Abstract

The general summation theorem for well-poised 5 F 4-series discovered by Dougall (Proc. Edinb. Math. Soc. 25:114–132, 1907) is shown to imply several infinite series of Ramanujan-type for 1/π and 1/π 2, including those due to Bauer (J. Reine Angew. Math. 56:101–121, 1859) and Glaisher (Q. J. Math. 37:173–198, 1905) as well as some recent ones by Levrie (Ramanujan J. 22:221–230, 2010).

Keywords

Dougall’s summation theorem for 5F4-series π-series of Ramanujan-type 

Mathematics Subject Classification (2000)

33C20 40A25 65B10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935) MATHGoogle Scholar
  2. 2.
    Baruah, N.D., Berndt, B.C.: Ramanujan’s Eisenstein series and new hypergeometric-like series for 1/π 2. J. Approx. Theory 160(1–2), 135–153 (2009) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Baruah, N.D., Berndt, B.C., Chan, H.H.: Ramanujan’s series for 1/π: a survey. Am. Math. Mon. 116, 567–587 (2009) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bauer, G.: Von den Coefficienten der Reihen von Kugelfunctionen einer Variabeln. J. Reine Angew. Math. 56, 101–121 (1859) MATHCrossRefGoogle Scholar
  5. 5.
    Berndt, B.C.: Ramanujan’s Notebooks (Part IV). Springer, New York (1994) (xii+451 pp.) MATHGoogle Scholar
  6. 6.
    Borwein, J.M., Borwein, P.B.: π & the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley, New York (1987) MATHGoogle Scholar
  7. 7.
    Borwein, J.M., Borwein, P.B.: More Ramanujan-type series for 1/π. In: Andrews, G.E., Berndt, B.C., Rankin, R.A. (eds.) Ramanujan Revisited. Proceedings of the Centenary Conference (Urbana-Champaign, 1987), pp. 359–374. Academic Press, San Diego (1988) Google Scholar
  8. 8.
    Borwein, J.M., Borwein, P.B.: Class number three Ramanujan type series for 1/π. J. Comput. Appl. Math. 46, 281–290 (1993) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chudnovsky, V., Chudnovsky, G.V.: Approximations and complex multiplication according to Ramanujan. In: Andrews, G.E., Berndt, B.C., Rankin, R.A. (eds.) Ramanujan Revisited, Proceedings of the Centenary Conference (Urbana-Champaign, 1987), pp. 375–472. Academic Press, San Diego (1988) Google Scholar
  10. 10.
    Dougall, J.: On Vandermonde’s theorem and some more general expansions. Proc. Edinb. Math. Soc. 25, 114–132 (1907) MATHCrossRefGoogle Scholar
  11. 11.
    Glaisher, J.W.L.: On series for 1/π and 1/π 2. Q. J. Math. 37, 173–198 (1905) MATHGoogle Scholar
  12. 12.
    Guillera, J.: Some binomial series obtained by the WZ-method. Adv. Appl. Math. 29(4), 599–603 (2002) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Guillera, J.: About a new kind of Ramanujan-type series. Exp. Math. 12(4), 507–510 (2003) MathSciNetMATHGoogle Scholar
  14. 14.
    Guillera, J.: Generators of some Ramanujan formulas. Ramanujan J. 11(1), 41–48 (2006) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Guillera, J.: Hypergeometric identities for 10 extended Ramanujan-type series. Ramanujan J. 15(2), 219–234 (2008) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Hardy, G.H.: Some formulae of Ramanujan. Proc. Lond. Math. Soc. 22, xii–xiii (1924) Google Scholar
  17. 17.
    Levrie, P.: Using Fourier–Legendre expansions to derive series for \(\frac{1}{\pi}\) and \(\frac{1}{\pi^{2}}\). Ramanujan J. 22, 221–230 (2010) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Ramanujan, S.: Modular equations and approximations to π. Q. J. Pure Appl. Math. 45, 350–372 (1914) Google Scholar
  19. 19.
    Zudilin, W.: Ramanujan-type formulae for 1/π: a second wind? In: Modular Forms and String Duality, pp. 179–188. Am. Math. Soc., Providence (2008) Google Scholar
  20. 20.
    Zudilin, W.: More Ramanujan-type formulas for 1/π 2. Usp. Mat. Nauk. 62, 211–212 (2007) (in Russian). Translation in Russ. Math. Surv. 62(3), 634–636 (2007) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institute of Combinatorial MathematicsHangzhou Normal UniversityHangzhouP.R. China
  2. 2.Dipartimento di MatematicaUniversità del SalentoLecceItaly

Personalised recommendations