Skip to main content

Circular Summation

  • Chapter
Book cover Ramanujan's Lost Notebook

Abstract

On page 54 in his lost notebook, Ramanujan derives identities for the sum of the nth powers of n general theta functions. He states a beautiful general theorem, and then provides five particular examples.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan’s theta function, Proc. Amer. Math. Soc. 128 (2000), 1333–1338.

    Article  MathSciNet  MATH  Google Scholar 

  2. G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Part I, Springer, New York, 2005.

    Google Scholar 

  3. G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Part II, Springer, New York, 2009.

    MATH  Google Scholar 

  4. A. Berkovich and H. Yesilyurt, Ramanujan’s identities and representation of integers by certain binary and quaternary quadratic forms, Ramanujan J. 20 (2009), 375–408.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Berkovich, F.G. Garvan, and H. Yesilyurt, manuscript in preparation.

    Google Scholar 

  6. B.C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.

    Book  MATH  Google Scholar 

  7. B.C. Berndt, On a certain theta-function in a letter of Ramanujan from Fitzroy House, Ganita 43 (1992), 33–43.

    MathSciNet  MATH  Google Scholar 

  8. B. C. Berndt, Ramanujan’s Notebooks, Part IV, Springer-Verlag, New York, 1994.

    Book  MATH  Google Scholar 

  9. B.C. Berndt, Ramanujan’s Notebooks, Part V, Springer-Verlag, New York, 1998.

    Book  MATH  Google Scholar 

  10. J.M. Borwein and P.B. Borwein, A cubic counterpart of Jacobi’s identity and the AGM, Trans. Amer. Math. Soc. 323 (1991), 691–701.

    MathSciNet  MATH  Google Scholar 

  11. H.H. Chan, Z.-G. Liu, and S.T. Ng, Circular summation of theta functions in Ramanujan’s lost notebook, J. Math. Anal. Appl. 316 (2006), 628–641.

    Article  MathSciNet  MATH  Google Scholar 

  12. S.H. Chan and Z.-G. Liu, On a new circular summation of theta functions, J. Number Thy. 130 (2010), 1190–1196.

    Article  MathSciNet  MATH  Google Scholar 

  13. K.S. Chua, Circular summation of the 13th powers of Ramanujan’s theta function, Ramanujan J. 5 (2001), 353–354.

    Article  MathSciNet  MATH  Google Scholar 

  14. K.S. Chua, The root lattice \(A_{n}^{*}\) and Ramanujan’s circular summation of theta functions, Proc. Amer. Math. Soc. 130 (2002), 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  15. T. Dai and X. Ma, A bivariate circular summation formula for cubic theta functions and its implications, J. Ramanujan Math. Soc. 26 (2011), 237–259.

    MathSciNet  MATH  Google Scholar 

  16. T. Murayama, On an identity of theta functions obtained from weight enumerators of linear codes, Proc. Japan Acad., Ser. A 74 (1998), 98–100.

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Ono, On the circular summation of the eleventh powers of Ramanujan’s theta functions, J. Number Thy. 76 (1999), 62–65.

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  19. S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.

    MATH  Google Scholar 

  20. S.S. Rangachari, On a result of Ramanujan on theta functions, J. Number Thy. 48 (1994), 364–372.

    Article  MathSciNet  MATH  Google Scholar 

  21. S.H. Son, Septic theta function identities in Ramanujan’s lost notebook, Acta Arith. 98 (2001), 361–374.

    Article  MathSciNet  MATH  Google Scholar 

  22. S.H. Son, Circular summations of theta functions in Ramanujan’s lost notebook, Ramanujan J. 8 (2004), 235–272.

    Article  MathSciNet  MATH  Google Scholar 

  23. S.H. Son, Ramanujan’s symmetric theta functions in his lost notebook, in Special Functions and Orthogonal Polynomials, D. Dominici and R.S. Maier, eds., Contemp. Math. 471, American Mathematical Society, Providence, RI, 2008, pp. 187–202.

    Chapter  Google Scholar 

  24. E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1966.

    Google Scholar 

  25. P. Xu, An elementary proof of Ramanujan’s circular summation formula and its generalizations, Ramanujan J. 27 (2012), 409–417.

    Article  MathSciNet  MATH  Google Scholar 

  26. X.-F. Zeng, A generalized circular summation of theta function and its applications, J. Math. Anal. Appl. 356 (2009), 698–703.

    Article  MathSciNet  MATH  Google Scholar 

  27. J.-M. Zhu, An alternate circular summation formula of theta functions and its applications, Applic. Anal. Discrete Math. 6 (2012), 114–125.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to George E. Andrews .

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media New York

About this chapter

Cite this chapter

Andrews, G.E., Berndt, B.C. (2012). Circular Summation. In: Ramanujan's Lost Notebook. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3810-6_9

Download citation

Publish with us

Policies and ethics