Skip to main content
SpringerLink
Log in

On some continued fraction expansions of the Rogers–Ramanujan type

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

By guessing the relative quantities and proving the recursive relation, we present some continued fraction expansions of the Rogers–Ramanujan type. Meanwhile, we also give some J-fraction expansions for the q-tangent and q-cotangent functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andrews, G.E.: On q-difference equations for certain well-poised basic hypergeometric series. Q. J. Math. (Oxford) 19, 433–447 (1968)

    Article  MATH  Google Scholar 

  2. Andrews, G.E.: An introduction to Ramanujan’s “lost” notebook. Am. Math. Mon. 86, 89–108 (1979)

    Article  MATH  Google Scholar 

  3. Andrews, G.E.: Ramunujan’s “lost” notebook III. The Rogers-Ramanujan continued fraction. Adv. Math. 41, 186–208 (1981)

    Article  MATH  Google Scholar 

  4. Andrews, G.E., Berndt, B.C., Jacobsen, L., Lamphere, R.L.: The Continued Fractions Found in the Unorganized Portions of Ramanujan’s Notebooks. Memoirs Amer. Math. Soc., vol. 477. Am. Math. Soc., Providence (1992)

    Google Scholar 

  5. Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook, Part I. Springer, New York (2005)

    Google Scholar 

  6. Berndt, B.C.: Ramanujan’s Notebooks, Part V. Springer, New York (1998)

    Book  MATH  Google Scholar 

  7. Bhargava, S., Adiga, C.: On some continued fraction identities of Srinivasa Ramanujan. Proc. Am. Math. Soc. 92, 13–18 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bowman, D., Mc Laughlin, J., Wyshinski, N.J.: A q-continued fraction. Int. J. Number Theory 2, 523–547 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bowman, D., Mc Laughlin, J., Wyshinski, N.J.: Continued fraction proofs of m-versions of some identities of Rogers-Ramanujan-Slater type. Ramanujan J. 25, 203–227 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chan, H.H., Huang, S.-S.: On the Ramanujan-Gordon-Göllnitz continued fraction. Ramanujan J. 1, 75–90 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fulmek, M.: A continued fraction expansion for a q-tangent function. Sémin. Lothar. Comb. 45 [B45b] (2000)

    MathSciNet  Google Scholar 

  12. Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  13. Gordon, B.: Some continued fractions of the Rogers-Ramanujan type. Duke Math. J. 32, 741–748 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hirschhorn, M.D.: A continued fraction. Duke Math. J. 41, 27–33 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hirschhorn, M.D.: Developments in Theory of Partitions. PhD Thesis, University of New South Wales (1980)

  16. Hirschhorn, M.D.: A continued fraction of Ramanujan. J. Aust. Math. Soc. A 29, 80–86 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jackson, F.H.: A basic-sine and cosine with symbolic solutions of certain differential equations. Proc. Edinb. Math. Soc. 22, 28–39 (1904)

    Article  MATH  Google Scholar 

  18. Jackson, F.H.: Examples of a generalization of Euler’s transformation for power series. Messenger Math. 57, 169–187 (1928)

    Google Scholar 

  19. Jones, W.B., Thron, W.J.: Continued Fractions. Analytic Theory and Applications. Addison-Wesley, Reading (1980)

    Google Scholar 

  20. Khovanskii, A.N.: The Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory. Noordhoff, Groningen (1963)

    Google Scholar 

  21. Lebesgue, V.A.: Sommation de quelques séries. J. Math. Pures Appl. 5, 42–71 (1840)

    Google Scholar 

  22. Matala-aho, T.: On the values of continued fractions: q-series. J. Approx. Theory 124, 139–153 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Matala-aho, T., Meril, V.: On the values of continued fractions: q-series. II. Int. J. Number Theory 2, 417–430 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mc Laughlin, J., Sills, A., Zimmer, P.: Rogers-Ramanujan computer searches. J. Symb. Comput. 44, 1068–1078 (2009)

    Article  MathSciNet  Google Scholar 

  25. Prodinger, H.: Combinatorics of geometrically distributed random variables: new q-tangent and q-secant numbers. Int. J. Math. Math. Sci. 24, 825–838 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Prodinger, H.: A continued fraction expansion for a q-tangent function: an elementary proof. Sémin. Lothar. Comb. 60 [B60b] (2008)

    MathSciNet  Google Scholar 

  27. Prodinger, H.: Continued fraction expansions for q-tangent and q-cotangent functions. Discrete Math. Theor. Comput. Sci. 12, 47–64 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Ramanathan, K.G.: Ramanujan’s continued fraction. Indian J. Pure Appl. Math. 16, 695–724 (1985)

    MathSciNet  MATH  Google Scholar 

  29. Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988)

    MATH  Google Scholar 

  30. Rogers, L.J.: Second memoir on the expansion of certain infinite products. Proc. Lond. Math. Soc. 25, 318–343 (1894)

    Article  Google Scholar 

  31. Selberg, A.: Über einige arithmetische Identitäten. Avh. - Nor. Vidensk.-Akad. Oslo, I Mat.-Naturv. Kl. 8, 3–23 (1936)

    Google Scholar 

  32. Selberg, A.: Collected Papers, vol. I. Springer, Berlin, (1989)

    Google Scholar 

  33. Shin, H., Zeng, J.: The q-tangent and q-secant numbers via continued fractions. Eur. J. Comb. 31, 1689–1705 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sills, A.: Finite Rogers-Ramanujan type identities. Electron. J. Comb. 10, R13 (2003)

    MathSciNet  Google Scholar 

  35. Slater, L.J.: Further identities of the Rogers-Ramanujan type. Proc. Lond. Math. Soc. 54, 147–167 (1952)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Combinatorics, LPMC-TJKLC, Nankai University, 300071, Tianjin, P.R. China

    Nancy S. S. Gu

  2. Department of Mathematics, University of Stellenbosch, 7602, Stellenbosch, South Africa

    Helmut Prodinger

Authors
  1. Nancy S. S. Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Helmut Prodinger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nancy S. S. Gu.

Additional information

N.S.S. Gu was supported by the National Natural Science Foundation of China, the PCSIRT Project of the Ministry of Education, and the Specialized Research Fund for the Doctoral Program of Higher Education of China (200800551042).

H. Prodinger was supported by International Science and Technology Agreement (Grant 67215).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gu, N.S.S., Prodinger, H. On some continued fraction expansions of the Rogers–Ramanujan type. Ramanujan J 26, 323–367 (2011). https://doi.org/10.1007/s11139-011-9329-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-011-9329-7

Keywords

Mathematics Subject Classification

Search

Navigation