Abstract
Srinivasa Ramanujan (1887–1920) recorded many modular equations in his notebooks. In this paper, we establish some new modular equations of signature two and three by using the theta-function identities of composite degrees.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adiga, C., Kim, T., Naika, M.S.M.: Modular equations in the theory of signature three and \(P\)-\(Q\) identities. Adv. Stud. Contemp. Math. 7, 33–40 (2003)
Adiga, C., Bulkhali, N.A.S., Ranganatha, D., Srivastava, H.M.: Some new modular relations for the Rogers–Ramanujan type functions of order eleven with applications to partitions. J. Number Theory 158, 281–297 (2016)
Adiga, C., Bulkhali, N.A.S., Simsek, Y., Srivastava, H.M.: A continued fraction of Ramanujan and some Ramanujan–Weber class invariants. Filomat 31, 3975–3997 (2017)
Berndt, B.C.: Ramanujan’s Notebooks. Part IV. Springer-Verlag, Berlin, Heidelberg and New York (1994)
Berndt, B.C.: Ramanujan’s Notebooks. Part V. Springer-Verlag, Berlin, Heidelberg and New York (1998)
Berndt, B.C., Zhang, L.-C.: Ramanujan’s identities for eta-functions. Math. Ann. 292, 561–573 (1992)
Naika, M.S.M.: A note on cubic modular equation of degree two. Tamsui Oxford J. Math. Sci 22, 1–8 (2006)
Naika, M.S.M.: \(P\)-\(Q\) eta function identities and computation of Ramanujan–Weber class invariants. J. Indian Math. Soc. ( New Ser.) 70, 121–134 (2003)
Naika, M.S.M., Chandan Kumar, S.: Some new Schläfli type modular equation of signature 4. Ramanujan Math. Soc. Lect. Notes Ser. No. 14, 185–199 (2010)
Ramanujan, S.: Notebooks, Vols. 1 and 2, Tata Institute of Fundamental Research, Bombay (1957)
Ramanujan, S.: The lost notebook and other unpublished papers. Narosa Publishing House, New Delhi (1988)
Saikia, N., Chetry, J.: Some new modular equations in Ramanujan’s alternate theory of signature 3. Ramanujan J. 50, 163–194 (2019)
Srivastava, H.M.: Operators of basic (or \(q\)-) calculus and fractional \(q\)-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. A 44, 327–344 (2020)
Srivastava, H.M., Chaudhary, M.P., Wakene, F.K.: A family of theta-function identities based upon \(q\)-binomial theorem and Heine’s transformations. Montes Taurus J. Pure Appl. Math. 2, 1–6 (2020)
Srivastava, H.M., Saikia, N.: Some congruences for over-partitions with restriction. Math. Notes 107, 488–498 (2020)
Srivastava, H.M., Srivastava, R., Chaudhary, M.P., Uddin, S.: A family of theta-function identities based upon combinatorial partition identities related to Jacobi’s triple-product identity, Mathematics, 8. Article ID 918, 1–14 (2020)
Srivatsa Kumar, B.R.: Shruthi: New modular equations of signature three in the spirit of Ramanujan. Filomat 34, 2847–2868 (2020)
Vasuki, K.R., Srivatsa Kumar, B.R.: Certain identities for Ramanujan–Göllnitz-Gordon continued fraction. J. Comput. Appl. Math. 187, 87–95 (2006)
Acknowledgements
The second-named author thanks SERB (Department of Science and Technology) of the Government of India under Project [EMR/2016/001601].
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no conflicts of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Srivastava, H.M., Kumar, B.R.S. & Narendra, R. Some modular equations analogous to Ramanujan’s identities. RACSAM 115, 59 (2021). https://doi.org/10.1007/s13398-021-01002-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-021-01002-w
Keywords
- Modular equations
- Theta functions
- Eta functions
- P-Q identities
- P-Q modular equations
- Class invariants and continued fractions