Skip to main content
Log in

Some new proofs of modular relations for the Göllnitz-Gordon functions

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

In this paper, we find new proofs of modular relations for the Göllnitz-Gordon functions established earlier by S.-S. Huang and S.-L. Chen. We use Schröter’s formulas and some simple theta-function identities of Ramanujan to establish the relations. We also find some new modular relations of the same nature.

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

Access this article

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. Baruah, N.D.: A few theta-function identities and some of Ramanujan’s modular equations. Ramanujan J. 4(3), 239–250 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)

    MATH  Google Scholar 

  3. Berndt, B.C., Choi, G., Choi, Y.S., Hahn, H., Yeap, B.P., Yee, A.J., Yesilyurt, H., Yi, J.: Ramanujan’s forty identities for the Rogers Ramanujan functions. Mem. Am. Math. Soc. 188(880), 1–96 (2007)

    MathSciNet  Google Scholar 

  4. Biagioli, A.J.F.: A proof of some identities of Ramanujan using modular forms. Glasg. Math. J. 31, 271–295 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bressoud, D.: Proof and generalization of certain identities conjectured by Ramanujan. Ph.D. Thesis, Temple University (1977)

  6. Chen, S.-L., Huang, S.-S.: New modular relations for the Göllnitz-Gordon functions. J. Number Theory 93, 58–75 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Darling, H.B.C.: Proofs of certain identities and congruences enunciated by S. Ramanujan. Proc. Lond. Math. Soc. (2) 19, 350–372 (1921)

    Article  Google Scholar 

  8. Göllnitz, H.: Partitionen mit Differenzenbedingungen. J. Reine Angew. Math. 225, 154–190 (1967)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Huang, S.-S.: On modular relations for Göllnitz-Gordon functions with application to partitions. J. Number Theory 68, 178–216 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ramanujan, S.: Proof of certain identities in combinatory analysis. Proc. Cambr. Philos. Soc. 19, 214–216 (1919)

    Google Scholar 

  12. Ramanujan, S.: Collected Papers. Cambridge University Press, Cambridge (1927). Reprinted by Chelsea, New York (1962). Reprinted by the American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  13. Ramanujan, S.: Notebooks (2 volumes). Tata Institute of Fundamental Research, Bombay (1957)

    Google Scholar 

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

    MATH  Google Scholar 

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

    Article  Google Scholar 

  16. Rogers, L.J.: On a type of modular relation. Proc. Lond. Math. Soc. 19, 387–397 (1921)

    Article  Google Scholar 

  17. Watson, G.N.: Proof of certain identities in combinatory analysis. J. Indian Math. Soc. 20, 57–69 (1933)

    Google Scholar 

  18. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1996). Indian edition is published by the Universal Book Stall, New Delhi (1991)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Nayandeep Deka Baruah.

Additional information

Research partially supported by grant SR/FTP/MA-02/2002 from DST, Govt. of India.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Baruah, N.D., Bora, J. & Saikia, N. Some new proofs of modular relations for the Göllnitz-Gordon functions. Ramanujan J 15, 281–301 (2008). https://doi.org/10.1007/s11139-007-9079-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-007-9079-8

Keywords

Mathematics Subject Classification (2000)

Navigation