S. Ramanujan documented several modular equations of degrees in his notebooks. These identities are used to evaluate Weber's class in variants, continued fractions and many more. In the present work, we establish modular equations of composite degrees using the known identities.
Citation: D. Anu Radha, B. R. Srivatsa Kumar. Some Schl$ \ddot{a} $fli type modular equations of composite degrees[J]. AIMS Mathematics, 2023, 8(3): 6167-6175. doi: 10.3934/math.2023311
S. Ramanujan documented several modular equations of degrees in his notebooks. These identities are used to evaluate Weber's class in variants, continued fractions and many more. In the present work, we establish modular equations of composite degrees using the known identities.
[1] | C. Adiga, T. Kim, M. S. Mahadeva Naika, H. S. Madhusudhan, On Ramanujan's cubic continued fraction and explicit evaluations of theta-functions, Indian J. Pure Appl. Math., 35 (2004), 1047–1062. |
[2] | C. Adiga, N. A. S. Bulkhali, D. Ranganatha, H. M. Srivatsa, Some new modular relations for the Rogers-Ramanujan type functions of order eleven with applications to partitions, J. Number Theory, 158 (2016), 281–297. https://doi.org/10.1016/j.jnt.2015.06.019 doi: 10.1016/j.jnt.2015.06.019 |
[3] | B. C. Berndt, Ramanujan's notebooks, Part III, Springer, New York, 1991. |
[4] | B. C. Berndt, Ramanujan's notebooks, Part IV, Springer, New York, 1996. |
[5] | M. Hanna, The modular equations, Proc. London Math. Soc., 28 (1928), 46–52. https://doi.org/10.1112/plms/s2-28.1.46 doi: 10.1112/plms/s2-28.1.46 |
[6] | M. S. M. Naika, A note on cubic modular equations of degree two, Tamsui Oxf, J. Math. Sci., 22 (2006), 1–8. |
[7] | S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957. |
[8] | S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 1988. |
[9] | N. Saikia, J. Chetry, Some new modular equations in Ramanujan's alternate theory of signature 3, Ramanujan J., 50 (2019), 163–194. https://doi.org/10.1007/s11139-018-0115-7 doi: 10.1007/s11139-018-0115-7 |
[10] | H. M. Srivastava, M. P. Chaudhary, F. K. Wakene, A family of theta-function identities based upon $q$-binomial theorem and Heine's transformations, Montes Taurus J. Pure Appl. Math., 2 (2020), 1–6. |
[11] | B. R. S. Kumar, Shruthi, New modular equations of signature three in the spirit of Ramanujan, Filomat, 34 (2020), 2847–2868. https://doi.org/10.2298/FIL2009847S doi: 10.2298/FIL2009847S |
[12] | D. Anuradha, B. R. S. Kumar, Some identities on modular equations of degree 5, Publ. Inst. Math., 111 (2022), 101–110. https://doi.org/10.2298/PIM2225101A doi: 10.2298/PIM2225101A |
[13] | B. R. S. Kumar, A. K. Rathie, N. V. U. Sayinath, Shruthi, A note on modular equations of signature 2 and their evaluations, Commun. Korean Math. Soc., 37, (2022), 31–44. |
[14] | K. R. Vasuki, B. R. S. Kumar, A note on Ramanujan-Schl$\ddot{a}$fli type mixed modular equations, S. E. Asian J. Math. Math. Sci., 5 (2006), 51–67. |
[15] | K. R. Vasuki, B. R. S. Kumar, Certain identities for Ramanujan-G$\dot{o}$llnitz-Gordon continued fraction, J. Comp. Appl. Math., 187 (2006), 87–95. https://doi.org/10.1016/j.cam.2005.03.038 doi: 10.1016/j.cam.2005.03.038 |
[16] | K. R. Vasuki, A. A. A. Kahtan, On certain theta function identities analogous to Ramanujan's $P$-$Q$ eta function identities, Appl. Math., 2 (2011), 874–882. https://doi.org/10.4236/am.2011.27117 doi: 10.4236/am.2011.27117 |