Abstract
In his second notebook, Ramanujan recorded total of seven P–Q modular equations involving theta-function \(f(-q)\) with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained for higher orders. As a consequence, several values of quotients of theta-function are evaluated. The cubic singular modulus is evaluated at \(q=\exp (-2\pi \sqrt{n/3})\) for \(n\in \{5k, 1/5k, 5/k, k/5\}\), where \(k\in \{4,7,16\}\).
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Acknowledgements
The authors would like to thank the anonymous referee for valuable corrections and comments. First author would like to thank DST for financial support through project no. SR/S4/MS:739/11 and second author would like to thank UGC for Junior Research Fellowship, Ref. No. F.17-58/2008(SA-I).
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Mahadeva Naika, M.S., Hemanthkumar, B. & Chandankumar, S. New P–Q mixed modular equations and their applications. Ann Univ Ferrara 64, 407–425 (2018). https://doi.org/10.1007/s11565-017-0286-y
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DOI: https://doi.org/10.1007/s11565-017-0286-y