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Eisenstein series and Ramanujan-type series for 1/π

Abstract

Using certain representations for Eisenstein series, we uniformly derive several Ramanujan-type series for 1/π.

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References

  1. 1.

    Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook, Part II. Springer, New York (2009)

    MATH  Google Scholar 

  2. 2.

    Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, London (1935)

    MATH  Google Scholar 

  3. 3.

    Baruah, N.D., Berndt, B.C.: Ramanujan’s series for 1/π arising from his cubic and quartic theories of elliptic functions. J. Math. Anal. Appl. 341, 357–371 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  4. 4.

    Bauer, G.: Von den Coefficienten der Reihen von Kugelfunctionen einer Variabeln. J. Reine Angew. Math. 56, 101–121 (1859)

    MATH  Google Scholar 

  5. 5.

    Berndt, B.C.: Ramanujan’s Notebooks, Part II. Springer, New York (1989)

    MATH  Google Scholar 

  6. 6.

    Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)

    MATH  Google Scholar 

  7. 7.

    Berndt, B.C.: Ramanujan’s Notebooks, Part V. Springer, New York (1998)

    MATH  Google Scholar 

  8. 8.

    Berndt, B.C., Chan, H.H.: Notes on Ramanujan’s singular moduli. In: Gupta, R., Williams, K.S. (eds.) Number Theory, pp. 7–16. Fifth Conference of the Canadian Number Theory Association. American Mathematical Society, Providence (1999)

    Google Scholar 

  9. 9.

    Berndt, B.C., Chan, H.H.: Eisenstein series and approximations to π. Ill. J. Math. 45, 75–90 (2001)

    MATH  MathSciNet  Google Scholar 

  10. 10.

    Berndt, B.C., Rankin, R.A.: Ramanujan: Letters and Commentary. American Mathematical Society/London Mathematical Society, Providence/London (1995)

    MATH  Google Scholar 

  11. 11.

    Berndt, B.C., Chan, H.H., Zhang, L.-C.: Ramanujan’s singular moduli. Ramanujan J. 1, 53–74 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  12. 12.

    Berndt, B.C., Chan, H.H., Liaw, W.-C.: On Ramanujan’s quartic theory of elliptic functions. J. Number Theory 88, 129–156 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  13. 13.

    Borwein, J.M., Borwein, P.B.: Pi and the AGM; A Study in Analytic Number Theory and Computational Complexity. Wiley, New York (1987)

    MATH  Google Scholar 

  14. 14.

    Borwein, J.M., Borwein, P.B.: Ramanujan’s rational and algebraic series for 1/π. J. Indian Math. Soc. 51, 147–160 (1987)

    MATH  MathSciNet  Google Scholar 

  15. 15.

    Borwein, J.M., Borwein, P.B.: More Ramanujan-type series for 1/π. In: Andrews, G.E., Askey, R.A., Berndt, B.C., Ramanathan, K.G., Rankin, R.A. (eds.) Ramanujan Revisited, pp. 359–374. Academic Press, Boston (1988)

    Google Scholar 

  16. 16.

    Borwein, J.M., Borwein, P.B.: Some observations on computer aided analysis. Not. Am. Math. Soc. 39, 825–829 (1992)

    Google Scholar 

  17. 17.

    Borwein, J.M., Borwein, P.B.: Class number three Ramanujan type series for 1/π. J. Comput. Appl. Math. 46, 281–290 (1993)

    MATH  Article  MathSciNet  Google Scholar 

  18. 18.

    Borwein, J.M., Borwein, P.B., Bailey, D.H.: Ramanujan, modular equations, and approximations to pi or how to compute one billion digits of pi. Am. Math. Mon. 96, 201–219 (1989)

    MATH  Article  MathSciNet  Google Scholar 

  19. 19.

    Chan, H.H., Liaw, W.-C.: Cubic modular equations and new Ramanujan-type series for 1/π. Pac. J. Math. 192, 219–238 (2000)

    MATH  Article  MathSciNet  Google Scholar 

  20. 20.

    Chan, H.H., Loo, K.P.: Ramanujan’s cubic continued fraction revisited. Acta Arith. 126, 305–313 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  21. 21.

    Chan, H.H., Verrill, H.: The Apéry numbers, the Almkvist-Zudilin numbers and new series for 1/π. Math. Res. Lett. 16, 405–420 (2009)

    MATH  MathSciNet  Google Scholar 

  22. 22.

    Chan, H.H., Liaw, W.-C., Tan, V.: Ramanujan’s class invariant λ n and a new class of series for 1/π. J. Lond. Math. Soc. 64(2), 93–106 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  23. 23.

    Chan, H.H., Chan, S.H., Liu, Z.: Domb’s numbers and Ramanujan-Sato type series for 1/π. Adv. Math. 186, 396–410 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  24. 24.

    Chowla, S.: Series for 1/K and 1/K 2. J. Lond. Math. Soc. 3, 9–12 (1928)

    Article  Google Scholar 

  25. 25.

    Chowla, S.: On the sum of a certain infinite series. Tôhoku Math. J. 29, 291–295 (1928)

    MATH  Google Scholar 

  26. 26.

    Chowla, S.: The Collected Papers of Sarvadaman Chowla, vol. 1. Les Publications Centre de Recherches Mathématiques, Montreal (1999)

    Google Scholar 

  27. 27.

    Chudnovsky, D.V., Chudnovsky, G.V.: Approximation and complex multiplication according to Ramanujan. In: Andrews, G.E., Askey, R.A., Berndt, B.C., Ramanathan, K.G., Rankin, R.A. (eds.) Ramanujan Revisited, pp. 375–472. Academic Press, Boston (1988)

    Google Scholar 

  28. 28.

    Guillera, J.: Some binomial series obtained by the WZ-method. Adv. Appl. Math. 29, 599–603 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  29. 29.

    Guillera, J.: About a new kind of Ramanujan-type series. Exp. Math. 12, 507–510 (2003)

    MATH  MathSciNet  Google Scholar 

  30. 30.

    Guillera, J.: Generators of some Ramanujan formulas. Ramanujan J. 11, 41–48 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  31. 31.

    Guillera, J.: A new method to obtain series for 1/π and 1/π 2. Exp. Math. 15, 83–89 (2006)

    MATH  MathSciNet  Google Scholar 

  32. 32.

    Guillera, J.: Hypergeometric identities for 10 extended Ramanujan type series. Ramanujan J. 15, 219–234 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  33. 33.

    Hardy, G.H.: Some formulae of Ramanujan. Proc. Lond. Math. Soc. 22(2), xii–xiii (1924)

    MathSciNet  Google Scholar 

  34. 34.

    Hardy, G.H.: Ramanujan. Cambridge University Press, Cambridge (1940); reprinted by Chelsea, New York (1960); reprinted by the American Mathematical Society, Providence (1999)

    Google Scholar 

  35. 35.

    Hardy, G.H.: Collected Papers, vol. 4. Clarendon Press, Oxford (1969)

    Google Scholar 

  36. 36.

    Ramanujan, S.: Modular equations and approximations to π. Quart. J. Math. (Oxford) 45, 350–372 (1914)

    MATH  Google Scholar 

  37. 37.

    Ramanujan, S.: Collected Papers. Cambridge University Press, Cambridge (1927); reprinted by Chelsea, New York (1962); reprinted by the American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  38. 38.

    Ramanujan, S.: Notebooks (2 volumes). Tata Institute of Fundamental Research, Bombay (1957)

    Google Scholar 

  39. 39.

    Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988)

    MATH  Google Scholar 

  40. 40.

    Zudilin, W.: Ramanujan-type formulae and irrationality measures of certain multiples of π. Mat. Sb. 196, 51–66 (2005)

    MathSciNet  Google Scholar 

  41. 41.

    Zudilin, W.: Quadratic transformations and Guillera’s formulae for 1/π 2. Math. Zametki 81, 335–340 (2007) (Russian); Math. Notes 81, 297–301 (2007)

    MathSciNet  Google Scholar 

  42. 42.

    Zudilin, W.: Ramanujan-type formulae for 1/π: A second wind? In: Yui, N., Verrill, H., Doran, C.F. (eds.) Modular Forms and String Duality. Fields Institute Communications, vol. 54, pp. 179–188. American Mathematical Society & The Fields Institute for Research in Mathematical Sciences, Providence (2008)

    Google Scholar 

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Correspondence to Bruce C. Berndt.

Additional information

Dedicated to George Andrews on the occasion of his 70-th birthday

N.D. Baruah research partially supported by BOYSCAST Fellowship grant SR/BY/M-03/05 from DST, Govt. of India.

B.C. Berndt research partially supported by NSA grant MSPF-03IG-124.

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Baruah, N.D., Berndt, B.C. Eisenstein series and Ramanujan-type series for 1/π . Ramanujan J 23, 17–44 (2010). https://doi.org/10.1007/s11139-008-9155-8

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Keywords

  • Eisenstein series
  • Theta-functions
  • Modular equations

Mathematics Subject Classification (2000)

  • 33C05
  • 33E05
  • 11F11
  • 11R29