The Ramanujan Journal

, 20:311 | Cite as

On Ramanujan’s function k(q)=r(q)r2(q2)

Article

Abstract

In both his second and lost notebooks, Ramanujan introduced a function, related to the Rogers–Ramanujan continued fraction and its quadratic transformation, and listed several of its properties. We extend these results and develop a systematic theory.

Keywords

Atkin-Lehner involution Eisenstein series Eta-function Ramanujan’s lost notebook Ramanujan’s second notebook Rogers-Ramanujan continued fraction Theta function 

Mathematics Subject Classification (2000)

11F11 11F20 14K25 33D52 33E05 

References

  1. 1.
    Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook, Part I. Springer, New York (2005) Google Scholar
  2. 2.
    Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991) MATHGoogle Scholar
  3. 3.
    Berndt, B.C.: Ramanujan’s Notebooks, Part V. Springer, New York (1998) MATHGoogle Scholar
  4. 4.
    Berndt, B.C.: Number Theory in the Spirit of Ramanujan. American Mathematical Society, Providence (2006) MATHGoogle Scholar
  5. 5.
    Blecksmith, R., Brillhart, J., Gerst, I.: Some infinite product identities. Math. Comput. 51, 301–314 (1988) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chan, H.H.: On the equivalence of Ramanujan’s partition identities and a connection with the Rogers–Ramanujan continued fraction. J. Math. Anal. Appl. 198, 111–120 (1996) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    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) Google Scholar
  8. 8.
    Chan, H.H., Lang, M.L.: Ramanujan’s modular equations and Atkin–Lehner involutions. Israel J. Math. 103, 1–16 (1998) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    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) MATHMathSciNetGoogle Scholar
  10. 10.
    Chan, H.H., Zudilin, W.: New representations for Apéry-like sequences. Preprint Google Scholar
  11. 11.
    Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc. 11, 308–339 (1979) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Cooper, S.: Inversion formulas for elliptic functions. Proc. Lond. Math. Soc. 99, 461–483 (2009) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Cooper, S., Hirschhorn, M.D.: On some infinite product identities. Rocky Mt. J. Math. 31, 131–139 (2001) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Cooper, S., Hirschhorn, M.D.: On some sum-to-product identities. Bull. Aust. Math. Soc. 63, 353–365 (2001) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Cooper, S., Toh, P.C.: Quintic and septic Eisenstein series. Ramanujan J. 19, 163–181 (2009) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Dobbie, J.M.: A simple proof of some partition formulae of Ramanujan’s. Q. J. Math. Oxf. (2) 6, 193–196 (1955) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Duke, W.: Continued fractions and modular functions. Bull. Am. Math. Soc., New Ser. 42, 137–162 (2005) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Gugg, C.: Two modular equations for squares of the Rogers–Ramanujan functions with applications. Ramanujan J. 18, 183–207 (2009) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Hirschhorn, M.: A simple proof of an identity of Ramanujan. J. Aust. Math. Soc., Ser. A 34, 31–35 (1983) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kang, S.Y.: Some theorems on the Rogers–Ramanujan continued fraction and associated theta function identities in Ramanujan’s lost notebook. Ramanujan J. 3, 91–111 (1999) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    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
  22. 22.
    Ramanujan, S.: Notebooks, 2 vols. Tata Institute of Fundamental Research, Bombay (1957) Google Scholar
  23. 23.
    Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988) MATHGoogle Scholar
  24. 24.
    Shen, L.-C.: On the additive formulae of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5. Trans. Am. Math. Soc. 345, 323–345 (1994) MATHCrossRefGoogle Scholar
  25. 25.
    Zagier, D.: Elliptic modular forms and their applications. In: Ranestad, K. (ed.) The 1-2-3 of Modular Forms. Universitext, pp. 1–103. Springer, Berlin (2008) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.IIMSMassey University-AlbanyAucklandNew Zealand

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