The Ramanujan Journal

, Volume 4, Issue 1, pp 43–50 | Cite as

The Borweins' Cubic Theta Function Identity and Some Cubic Modular Identities of Ramanujan

  • Zhi-Guo Liu


In this paper the author will give new proofs of the Borweins' cubic theta function identity and a related identity relying on the properties of elliptic functions and the technique of comparing constant terms.

theta functions elliptic functions modular equations modular forms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B.C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, N.Y., 1991.Google Scholar
  2. 2.
    B.C. Berndt, S. Bhargava, and F.G. Garvan, “Ramanujan's theories of elliptic functions to alternative bases,” Trans. Amer. Math. Soc. 347 (1995) 4136–4244.Google Scholar
  3. 3.
    J.M. Borwein and P.B. Borwein, “A cubic counterpart of Jacobi's identity and AGM,” Trans. Amer. Math. Soc. 323 (1991) 691–701.Google Scholar
  4. 4.
    J.M. Borwein, P.B. Borwein, and F.G. Garvan, “Some cubic modular identities of Ramanujan,” Trans. Amer. Math. Soc. 343 (1994) 35–47.Google Scholar
  5. 5.
    M. Hirschhorn, F. Garvan, and J.M. Borwein, “Cubic analogues of the Jacobian theta function θ(z, q),” Canad. J. Math. 45 (1993) 673–694.Google Scholar
  6. 6.
    S. Ramanjan, Notebooks, 2 vols., Tata Institute of Fundamental Research, Bombay, 1957.Google Scholar
  7. 7.
    E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th edn., Cambridge Univ. Press, Cambridge, 1966.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Zhi-Guo Liu
    • 1
  1. 1.Xinxing Education College, XinxiangHenanP.R. China

Personalised recommendations