Abstract
In this paper, we prove some new modular equations in the theory of signature 3 or cubic modular equations by using theta-function identities. Particularly, we prove modular equations of prime degrees 2, 3, 5, 7 and 11. We also prove some modular equations of composite degrees \(\{1, 2, 4\}, \{1, 3, 9\}, \{1, 7, 49\}, \{1, 2, 3, 6\}, \{1, 2, 5, 10\}, \{1, 2, 7, 14\}, \{1, 2, 11, 22\}, \{1, 2, 13, 26\} \{1, 3, 4, 12\}, \{1, 3, 5, 15\}, \hbox { and } \{1, 3, 7, 21\}\).
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Adiga, C., Kim, T., Naika, M.S.M.: Modular equations in the theory of signature 3 and \(P-Q\) identities. Adv. Stud. Contemp. Math. 7, 33–40 (2003)
Adiga, C., Kim, T., Naika, M.S.M., Madhusudhan, H.S.: On Ramanujan’s cubic continued fraction and explicit evaluations of theta-functions. Indian J. Pure Appl. Math. 35(9), 1047–1062 (2004)
Baruah, N.D.: Modular equations for Ramanujan’s cubic continued fraction. J. Math. Anal. Appl. 268, 244–255 (2002)
Baruah, N.D., Saikia, N.: Some general theorems on the explicit evaluations of Ramanujan’s cubic continued fraction. J. Comput. Appl. Math. 160, 37–51 (2003)
Baruah, N.D., Saikia, N.: Some new explicit values of Ramanuja’s continued fractions. Indian J. Math. 46, 197–222 (2004)
Baruah, N.D., Saikia, N.: Two parameters for Ramanujan’s theta-functions and their explicit values. Rocky Mt. J. Math. 37, 1747–1790 (2007)
Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)
Berndt, B.C.: Ramanujan’s Notebooks, Part IV. Springer, New York (1994)
Berndt, B.C.: Ramanujan’s Notebooks, Part V. Springer, New York (1998)
Berndt, B.C., Bhargava, S., Garvan, F.G.: Ramanujan’s theories of elliptic functions to alternative bases. Trans. Am. Math. Soc. 347, 4163–4244 (1995)
Borwein, J.M., Bowein, P.B.: A cubic counter part of Jacobi’s identity and the AGM. Trans. Am. Math. Soc. 323, 691–701 (1991)
Das, K.: New proofs of some modular equations in the theory of signature 3. J. Indian Math. Soc. 82(3–4), 23–37 (2015)
Naika, M.S.M.: A note on cubic modular equations of degree two. Tamsui Oxford J. Math. Sci. 22(1), 1–8 (2006)
Ramanujan, S.: Modular equations and approximations to \(\pi \). Q. J. Math. 45, 350–372 (1914)
Ramanujan, S.: Notebooks (2 Volumes). Tata Institute of Fundamental Research, Bombay (1957)
Ramanujan, S.: Collected Papers. Chelsa, NewYork (1962)
Saikia, N.: Modular identities and explicit values of a continued fraction of order twelve. JP J. Algebra Number Theory Appl. 22, 127–154 (2011)
Saikia, N.: Ramanujan’s modular equations and Weber-Ramanujan’s class invariants \(G_n\) and \(g_n\). Bull. Math. Sci. 2, 205–223 (2012)
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Saikia, N., Chetry, J. Some new modular equations in Ramanujan’s alternate theory of signature 3. Ramanujan J 50, 163–194 (2019). https://doi.org/10.1007/s11139-018-0115-7
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DOI: https://doi.org/10.1007/s11139-018-0115-7