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Theta functions and Eisenstein series

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Abstract

We define two elliptic functions and prove an identity connecting one function and a sum of Eisenstein series. We find a recurrence relation involving the Eisenstein series and Dedekind \(\eta \)-products. We show equivalence of two elliptic functions under imaginary transformation and express the other function as an Eisenstein series. Combining the identity for the elliptic functions a curious identity related to a continued fraction found in Ramanujan’s “lost” notebook is derived.

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References

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

    Google Scholar 

  2. Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory. Springer, New York (1976)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  4. Chan, H.H., Liu, Z.-G.: Elliptic functions to the quintic base. Pac. J. Math. 226, 53–64 (2006)

  5. Cooper, S., Lam, H.Y.: Eisenstein series and elliptic functions on \(\Gamma _0 (10),\). Adv. Appl. Math. 46, 192–208 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  7. Liu, Z.-G.: Some Eisenstein series identities related to modular equations of the seventh order. Pac. J. Math. 209, 103–130 (2003)

    Article  MATH  Google Scholar 

  8. Liu, Z.-G.: Two theta function identities and some Eisenstein series identities of Ramanujan. Rocky Mt. J. Math. 34, 713–732 (2004)

  9. Liu, Z.-G.: A three-term theta function identity and its application. Adv. Math. 195, 1–23 (2006)

  10. Liu, Z.-G.: A theta function identity and the Eisenstein series on \(\Gamma _0 (5)\). J. Ramanujan Math. Soc. 22, 283–298 (2007)

  11. Ramanujan, S.: Collected Papers. Cambridge University Press, Cambridge (1927). Reprinted by Chelsea New York 1960, reprinted by American Mathematical Society Providence (2000)

  12. Ramanujan, S.: The Lost Notebook and Other Unpublished Paper. Narosa, New Delhi and Springer, Berlin (1988)

  13. Srivastava, B.: A simple proof of identities of Legendre and Ramanujan. Bull. Math. Anal. Appl. 3(1), 8–13 (2011)

    MathSciNet  Google Scholar 

  14. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1966)

    Google Scholar 

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Acknowledgments

We are thankful to the referees for their comments.

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Authors and Affiliations

  1. Department of Mathematics and Astronomy, Lucknow University, Lucknow, India

    Bhaskar Srivastava

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  1. Bhaskar Srivastava
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Correspondence to Bhaskar Srivastava.

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Srivastava, B. Theta functions and Eisenstein series. Bol. Soc. Mat. Mex. 21, 189–203 (2015). https://doi.org/10.1007/s40590-014-0044-4

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  • DOI: https://doi.org/10.1007/s40590-014-0044-4

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