Abstract
On page 237–238 of his second notebook, Ramanujan recorded five modular equations of composite degree 25. Berndt proved all these using the method of parametrization. He also expressed that his proofs undoubtedly often stray from the path followed by Ramanujan. The purpose of this paper is to give direct proofs to four of the five modular equations using the identities known to Ramanujan.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alloush, K.A.A.: A study on q-series, continued fractions and modular equations motivated by the works of S. Ramanujan. Thesis submitted to the University of Mysore (2013)
Berndt, B.C.: Ramanujan Notebooks, Part III. Springer, New York (1991)
Berndt, B.C.: Ramanujan Notebooks, Part IV. Springer, New York (1994)
Bhargava, S., Vasuki, K.R., Rajanna, K.R.: On some Ramanujan identities for the ratios of eta functions. Ukr. Math. J. 66(8), 1131–1151 (2015)
Ramanujan, S.: Notebooks (2 Volumes). Tata Institute of Fundamental Research, Bombay (1957)
Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa Publishing House, New Delhi (1988)
Acknowledgements
The authors would like to thank the anonymous referee for many invaluable suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author is supported by Grant No. F. 510/12/DRS-II/2018 (SAP-I) by the University Grants Commission, India. The second author is supported by Grant No. 09/119(0221)/2019-EMR-1 by the funding agency CSIR, India, under CSIR-JRF.
Rights and permissions
About this article
Cite this article
Vasuki, K.R., Yathirajsharma, M.V. On modular equations of degree 25. Ramanujan J 56, 743–752 (2021). https://doi.org/10.1007/s11139-020-00276-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-020-00276-9