Ukrainian Mathematical Journal

, Volume 66, Issue 8, pp 1131–1151 | Cite as

On Some Ramanujan Identities for the Ratios of Eta-Functions

  • S. Bhargava
  • K. R. Vasuki
  • K. R. Rajanna

We give direct proofs of some of Ramanujan’s P-Q modular equations based on simply proved elementary identities from Chapter 16 of his Second Notebook.


Theta Function Require Result Geometric Series Double Series Modular Equation 
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  1. 1.
    C. Adiga, A Study of Some Identities Stated by Srinivasa Ramanujan in His “Lost” Note Book and Earlier Works, Doctoral Thesis, University of Mysore (1983).Google Scholar
  2. 2.
    C. Adiga, B. C. Berndt, S. Bhargava, and G. N.Watson, “Chapter 16 of Ramanujan’s second notebook: Theta functions and q-series,” Providence, American Mathematical Society, 53 (1985).Google Scholar
  3. 3.
    B. C. Berndt, Ramanujan’s Notebooks, Springer-Verlag, New York (1991), Pt III.CrossRefzbMATHGoogle Scholar
  4. 4.
    B. C. Berndt, Ramanujan’s Notebooks, Springer-Verlag, New York (1994), Pt IV.CrossRefzbMATHGoogle Scholar
  5. 5.
    B. C. Berndt, Ramanujan’s Notebooks, Springer-Verlag, New York (1998), Pt V.CrossRefzbMATHGoogle Scholar
  6. 6.
    B. C. Berndt, “Modular equations in Ramanujan’s lost notebook,” Number Theory, Eds. R. P. Bambah, V. C. Dumir, and R. J. Hans-Gill, Hindustan Book Agency, New Delhi (2000), pp. 55–74.CrossRefGoogle Scholar
  7. 7.
    B. C. Berndt and L.-C. Zhang, “Ramanujan’s identities for eta functions,” Math. Ann., 292, 561–573 (1992).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    S. Bhargava and C. Adiga, “Simple proofs of Jacobi’s two and four square theorems,” Int. J. Math. Ed. Sci. Tech., 19, 779–782 (1988).Google Scholar
  9. 9.
    G. H. Hardy, Ramanujan, 3 rd edn., Chelsea, New York (1978).Google Scholar
  10. 10.
    S. Y. Kang, “Some theorems on the Roger’s–Ramanujan continued fraction and associated theta function identities in Ramanujan’s lost notebook,” Ramanujan J., 3, 91–111 (1999).CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    S. Ramanujan, Notebooks, Tata Inst. Fundam. Res., Bombay (1957), Vols. 1, 2.Google Scholar
  12. 12.
    S. Ramanujan, The Lost Notebook and Other Unpublished Paper, Narosa, New Delhi (1988).Google Scholar
  13. 13.
    K. Venkatachaliengar, “Development of elliptic functions according to Ramanujan,” Tech. Rept, Madurai, Madurai Kamaraj Univ., 2 (1988).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • S. Bhargava
    • 1
  • K. R. Vasuki
    • 1
  • K. R. Rajanna
    • 2
  1. 1.University of MysoreMysoreIndia
  2. 2.MVJ College EngineeringBangaloreIndia

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