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On a Ramanujan’s Eisenstein series identity of level fifteen

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Abstract

On pages 255–256 of his second notebook, Ramanujan recorded an Eisenstein series identity of level 15 without offering a proof. Previously, Berndt (Ramanujan’s Notebooks: Part III (1991) (New York: Springer)) proved this identity using the theory of modular forms. In this paper, we give an elementary proof of this identity. In the process, we also give an elementary proof of three Ramanujan’s \(P-Q\) identities of level 15. Further, using the \(P-Q\) identities we prove four Ramanujan type Eisenstein series of level 15 due to Cooper and Ye (Trans. Am. Math. Soc. 368 (2016) 7883–7910), where they have proved using the theory of modular forms.

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References

  1. Baruah N D and Berndt B C, Eisenstein series and Ramanujan-type series for \(1/\pi \), Ramanujan J. 23 (2010) 17–44

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  3. Berndt B C, Ramanujan’s Notebooks: Part IV (1994) (New York: Springer)

    Book  Google Scholar 

  4. Berndt B C, Chan H H, Incomplete elliptic integrals in Ramanujan’s lost notebook, Contemp. Math. 254 (2000) 79–126

    Article  MathSciNet  Google Scholar 

  5. Bhargava S, Vasuki K R, Rajanna K R, On some Ramanujan identities for the ratios of eta-functions, Ukrainian Math. J. 66 (2015) 1131–1151

    Article  MathSciNet  Google Scholar 

  6. Cooper S, and Ye D, Level 14 and 15 analogues of Ramanujan’s elliptic functions to alternative bases, Trans. Am. Math. Soc. 368 (2016) 7883–7910

    Article  MathSciNet  Google Scholar 

  7. Ramanujan S, Modular equations and approximations to \(\pi \), Q. J. Math. 45 (1914) 350–372

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referee for the valuable comments.

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Authors and Affiliations

  1. Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysuru, 570 006, India

    E N Bhuvan & K R Vasuki

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  1. E N Bhuvan
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  2. K R Vasuki
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Correspondence to K R Vasuki.

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Communicating Editor: Sanoli Gun

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Bhuvan, E.N., Vasuki, K.R. On a Ramanujan’s Eisenstein series identity of level fifteen. Proc Math Sci 129, 57 (2019). https://doi.org/10.1007/s12044-019-0498-4

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  • DOI: https://doi.org/10.1007/s12044-019-0498-4

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