On the Generalized Rogers-Ramanujan Continued Fraction

  • Bruce C. Berndt
  • Ae Ja Yee
Part of the Developments in Mathematics book series (DEVM, volume 10)


On page 26 in his lost notebook, Ramanujan states an asymptotic formula for the generalized Rogers-Ramanujan continued fraction. This formula is proved and made slightly more precise. A second primary goal is to prove another continued fraction representation for the Rogers-Ramanujan continued fraction conjectured by R. Blecksmith and J. Brillhart. Two further entries in the lost notebook are examined. One of them is an identity bearing a superficial resemblance to the generating function for the generalized Rogers-Ramanujan continued fraction. Thus, our third main goal is to establish, with the help of an idea of E Franklin, a partition bijection to prove this identity.

Key words

Rogers-Ramanujan continued fraction generalized Rogers-Ramanujan continued fraction Franklin’s involution Ramanujan’s lost notebook 


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  1. 1.
    G.E. Andrews, The Theory of Partitions, Addison-Wesley, Reading. MA, 1976; reissued by Cambridge University Press, Cambridge. 1998.Google Scholar
  2. 2.
    B.C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.Google Scholar
  3. 3.
    B.C. Berndt, Ramanujan’s Notebooks, Part IV, Springer-Verlag, New York, 1994.Google Scholar
  4. 4.
    B.C. Berndt, Ramanujan’s Notebooks, Part V, Springer-Verlag, New York, 1998.Google Scholar
  5. 5.
    B.C. Berndt and J. Sohn, “Asymptotic formulas for two continued fractions in Ramanujan’s lost notebook,” J. London Math. Soc. (2) 65 (2002), 271–284.MathSciNetzbMATHGoogle Scholar
  6. 6.
    J. Brillhart, email to B.C. Berndt, January 27, 2002.Google Scholar
  7. 7.
    H.H. Chan, “On Ramanujan’s cubic continued fraction.’ Acta Arith. 7311995 ), 343–355.Google Scholar
  8. 8.
    S. Ramanujan, “Proof of certain identities in combinatory analysis,” Proc. Cambridge Philos. Soc. 19 (1919), 214–216.Google Scholar
  9. 9.
    S. Ramanujan, Collected Papers, Cambridge University Press. Cambridge. 1927; reprinted by Chelsea, New York, 1960; reprinted by the American Mathematical Society. Providence, RI. 2000.Google Scholar
  10. 10.
    S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.Google Scholar
  11. 11.
    S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.zbMATHGoogle Scholar
  12. 12.
    L.J. Rogers, “Second memoir on the expansion of certain infinite products,” Proc. London Math. Soc. 25 (1894), 318–343.Google Scholar
  13. 13.
    J. Sohn, email to B. C. Berndt, August 12, 2002.Google Scholar
  14. 14.
    J.J. Sylvester, Collected Papers, Vol. 4, Cambridge University Press, London, 1912; reprinted by Chelsea, New York. 1974.Google Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Bruce C. Berndt
    • 1
  • Ae Ja Yee
    • 2
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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