The Ramanujan Journal

, Volume 35, Issue 3, pp 391–403 | Cite as

On cubic multisections of Eisenstein series

  • Andrew Alaniz
  • Tim Huber


A systematic procedure for generating cubic multisections of Eisenstein series is given. The relevant series are determined from Fourier expansions for Eisenstein series by restricting the congruence class of the summation index modulo three. We prove that the resulting series are rational functions of η(τ) and η(3τ), where η is the Dedekind eta function. A more general treatment of cubic dissection formulas is given by describing the dissection operators in terms of linear transformations. These operators exhibit properties that mirror those of similarly defined quintic operators.


Eisenstein series Cubic theta functions Cubic multisections t-Cores 

Mathematics Subject Classification (2010)

11F11 11F33 


  1. 1.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999) CrossRefMATHGoogle Scholar
  2. 2.
    Bannai, E., Koike, M., Munemasa, A., Sekiguchi, J.: Some results on modular forms—subgroups of the modular group whose ring of modular forms is a polynomial ring. In: Groups and Combinatorics—in Memory of Michio Suzuki. Adv. Stud. Pure Math., vol. 32, pp. 245–254. Math. Soc. Japan, Tokyo (2001) Google Scholar
  3. 3.
    Berndt, B.C., Bhargava, S., Garvan, F.: Ramanujan’s theories of elliptic functions to alternative bases. Trans. Am. Math. Soc. 347(11), 4163–4244 (1995) MATHMathSciNetGoogle Scholar
  4. 4.
    Borwein, J.M., Borwein, P.B.: A cubic counterpart of Jacobi’s identity and the AGM. Trans. Am. Math. Soc. 323(2), 691–701 (1991) MATHGoogle Scholar
  5. 5.
    Borwein, J.M., Borwein, P.B., Garvan, F.G.: Some cubic modular identities of Ramanujan. Trans. Am. Math. Soc. 343(1), 35–47 (1994) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Chandrasekharan, K.: Elliptic Functions. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 281. Springer, Berlin (1985) MATHGoogle Scholar
  7. 7.
    Cooper, S.: Cubic elliptic functions. Ramanujan J. 11(3), 355–397 (2006) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Garvan, F., Kim, D., Stanton, D.: Cranks and t-cores. Invent. Math. 101(1), 1–17 (1990) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Granville, A., Ono, K.: Defect zero p-blocks for finite simple groups. Trans. Am. Math. Soc. 348(1), 331–347 (1996) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Huber, T.: A theory of theta functions to the quintic base. J. Number Theory 134, 49–92 (2014) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Jacobi, C.G.J.: Fundamenta nova theoriae functionum ellipticarum. Bornträger, Regiomonti (1829) Google Scholar
  12. 12.
    Köhler, G.: Eta Products and Theta Series Identities. Springer Monographs in Mathematics. Springer, Heidelberg (2011) CrossRefMATHGoogle Scholar
  13. 13.
    Lahiri, D.B.: Some congruences for the elementary divisor functions. Am. Math. Mon. 76, 395–397 (1969) CrossRefMATHGoogle Scholar
  14. 14.
    Ramanujan, S.: Notebooks, vols. 1, 2. Tata Institute of Fundamental Research, Bombay (1957) MATHGoogle Scholar
  15. 15.
    Robbins, N.: On t-core partitions. Fibonacci Q. 38(1), 39–48 (2000) MATHGoogle Scholar
  16. 16.
    Sebbar, A.: Modular subgroups, forms, curves and surfaces. Can. Math. Bull. 45(2), 294–308 (2002) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Shen, L.-C.: On some cubic modular identities. Proc. Am. Math. Soc. 119(1), 203–208 (1993) CrossRefMATHGoogle Scholar
  18. 18.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions: With an Account of the Principal Transcendental Functions, 4th edn. Cambridge University Press, New York (1962). Reprinted MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas, Pan AmericanEdinburgUSA

Personalised recommendations