Research article

Some Schl$ \ddot{a} $fli type modular equations of composite degrees

  • Received: 08 October 2022 Revised: 22 November 2022 Accepted: 01 December 2022 Published: 30 December 2022
  • MSC : Primary 11F03, 11F27, 14H42

  • 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

    Related Papers:

  • 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.



    加载中


    [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    [8] S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 1988.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
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1199) PDF downloads(82) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog