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Indian Journal of Pure and Applied Mathematics

, Volume 50, Issue 4, pp 1097–1105 | Cite as

Revisit to Ramanujan’s modular equations of degree 21

  • K. R. VasukiEmail author
  • E. N. BhuvanEmail author
  • T. AnushaEmail author
Article
  • 13 Downloads

Abstract

S. Ramanujan recorded six modular equations of degree 21 in his notebooks without recording proofs. B. C. Berndt proved all these modular equations by using the theory of modular forms. Recently Vasuki and Sharath proved two of them by using tools known to Ramanujan [5]. In this paper, we provide classical proof of remaining four identities.

Key words

Dedekind eta-function modular equation 

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Notes

Acknowledgement

The authors are grateful to the anonymous referees for their helpful comments and suggestions. T. Anusha is supported by grant No. 09/119(0202)/2017-EMR-1 by the funding agency Council of Scientific and Industrial Research (CSIR), India, under CSIR-JRF.

References

  1. 1.
    N. D. Baruah, On some of Ramanujan’s Schläfli-type “mixed” modular equations, J. Number Theory, 100(2) (2003), 270–294.MathSciNetCrossRefGoogle Scholar
  2. 2.
    B. C. Berndt, Ramanujan’s notebooks: Part III, Springer New York, (1991).CrossRefGoogle Scholar
  3. 3.
    B. C. Berndt, Ramanujan’s notebooks: Part IV, Springer New York, (1994).CrossRefGoogle Scholar
  4. 4.
    S. Bhargava, C. Adiga, and M. S. Mahadeva Naika, A new class of modular equations in Ramanujan’s alternative theory of elliptic functions of signature 4 and some new P-Q eta-function identities, Indian J. Math., 45(1) (2003), 23–39.MathSciNetzbMATHGoogle Scholar
  5. 5.
    S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research Bombay, (1957).zbMATHGoogle Scholar
  6. 6.
    M. Somos, Personal CommunicationGoogle Scholar
  7. 7.
    K. R. Vasuki and G. Sharath, On Ramanujan’s modular equations of degree 21, J. Number Theory, 133 (2013), 437–445.MathSciNetCrossRefGoogle Scholar
  8. 8.
    R. G. Veeresha, An elementary approach to Ramanujan’s modular equation of degree 7 and its appilications, Doctoral Thesis, University of Mysore, 2015.Google Scholar

Copyright information

© Indian National Science Academy 2019

Authors and Affiliations

  1. 1.Department of Studies in MathematicsUniversity of MysoreManasagangotri, MysuruIndia

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