The Ramanujan Journal

, Volume 19, Issue 2, pp 163–181 | Cite as

Quintic and septic Eisenstein series

  • Shaun CooperEmail author
  • Pee Choon Toh


Using results that were well-known to Ramanujan, we give proofs of some results for Eisenstein series in the lost notebook. Our proofs have the additional advantage that it is not necessary to know the results in advance; that is, the proofs are derivations as opposed to verifications.


Convolution sum Dedekind eta function Eisenstein series Lost notebook Modular function Rogers-Ramanujan continued fraction 

Mathematics Subject Classification (2000)

11F20 33E05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alaca, A., Alaca, S., Williams, K.S.: Evaluation of the convolution sums ∑+18m=n σ()σ(m) and ∑2+9m=n σ()σ(m). Int. Math. Forum 2, 45–68 (2007) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook. Part I. Springer, New York (2005) Google Scholar
  3. 3.
    Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991) zbMATHGoogle Scholar
  4. 4.
    Berndt, B.C., Chan, H.H., Sohn, J., Son, S.H.: Eisenstein series in Ramanujan’s lost notebook. Ramanujan J. 4, 81–114 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chan, H.H.: Triple product identity, quintuple product identity and Ramanujan’s differential equations for the classical Eisenstein series. Proc. Am. Math. Soc. 135, 1987–1992 (2007) zbMATHCrossRefGoogle Scholar
  6. 6.
    Chan, H.H., Cooper, S.: Powers of theta functions. Pac. J. Math. 235, 1–14 (2008) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Chan, H.H., Ong, Y.L.: On Eisenstein series and \(\sum\sp \infty\sb {m,n=-\infty}q\sp {m\sp 2+mn+2n\sp 2}\) . Proc. Am. Math. Soc. 127, 1735–1744 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dobbie, J.M.: A simple proof of some partition formulae of Ramanujan’s. Q. J. Math. Oxf. 6(2), 193–196 (1955) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Duke, W.: Continued fractions and modular functions. Bull. Am. Math. Soc. (New Ser.) 42, 137–162 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hirschhorn, M.: A simple proof of an identity of Ramanujan. J. Austral. Math. Soc. Ser. A 34, 31–35 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hirschhorn, M.: An identity of Ramanujan, and applications. In: Ismail, M.E.H., Stanton, D. (eds.) q-Series from a Contemporary Perspective, pp. 229–234. American Mathematical Society, Providence (2000) Google Scholar
  12. 12.
    Landau, E.: Elementary Number Theory. Chelsea, New York (1958) zbMATHGoogle Scholar
  13. 13.
    Lewis, R., Liu, Z.-G.: On two identities of Ramanujan. Ramanujan J. 3, 335–338 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Liu, Z.-G.: Some Eisenstein series identities related to modular equation of the seventh order. Pac. J. Math. 209, 103–130 (2003) zbMATHCrossRefGoogle Scholar
  15. 15.
    Liu, Z.-G.: Two theta function identities and some Eisenstein series identities of Ramanujan. Rocky M. J. Math. 34, 713–732 (2004) zbMATHCrossRefGoogle Scholar
  16. 16.
    Raghavan, S., Rangachari, S.S.: Ramanujan’s elliptic integrals and modular identities. In: Number Theory and Related Topics, pp. 119–149. Oxford University Press, Bombay (1989) Google Scholar
  17. 17.
    Ramanujan, S.: On certain arithmetical functions. Trans. Camb. Philos. Soc. 22, 159–184 (1916) Google Scholar
  18. 18.
    Ramanujan, S.: Some properties of p(n), the number of partitions of n. Proc. Camb. Philos. Soc. 19, 207–210 (1919) zbMATHGoogle Scholar
  19. 19.
    Ramanujan, S.: Notebooks (2 volumes). Tata Institute of Fundamental Research, Bombay (1957) Google Scholar
  20. 20.
    Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988) zbMATHGoogle Scholar
  21. 21.
    Royer, E.: Evaluating convolution sums of the divisor function by quasimodular forms. Int. J. Number Theory 3, 231–261 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1927) zbMATHGoogle Scholar
  23. 23.
    Williams, K.S.: On a double series of Chan and Ong. Georgian Math. J. 13, 793–805 (2006) zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institute of Information and Mathematical SciencesMassey University-AlbanyAucklandNew Zealand
  2. 2.Department of MathematicsNational University of SingaporeSingaporeSingapore

Personalised recommendations