Theorems on analogous of Ramanujan’s remarkable product of theta-function and their explicit evaluations
Resumo
In this article, we define Em,n for any positive real numbers m and n involving Ramanujan’s product of theta-functions ψ(−q) and f(q), which is analogous to Ramanujan’s remarkable product of theta-functions and establish its several properties by Ramanujan. We establish general theorems for the explicit evaluations of Em,n and its explicit values.
Downloads
Referências
C. Adiga, Taekyun Kim, M. S. Mahadeva Naika and H. S. Madhusudhan, On Ramnujan's cubic continued fraction and explicit evaluations of theta-functions, Indian J. pure appl. math., 35(9) (2004), 1047-1062.
B. C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0965-2
B. C. Berndt, Ramanujan's Notebooks, Part IV, Springer-Verlag, New York, 1994. https://doi.org/10.1007/978-1-4612-0879-2
B. C. Berndt, Ramanujan's Notebooks, Part V, Springer-Verlag, New York, 1997.
B. C. Berndt, H. H. Chan and L.-C. Zhang, Ramanujan's remarkable product of the theta-function, Proc. Edinburgh Math. Soc., 40 (1997), 583-612 . https://doi.org/10.1017/S0013091500024032
M. S. Mahadeva Naika, Some theorems on Ramanujan's cubic continued fraction and related identities. Tamsui Oxf. J. Math. Sci. 24(3) (2008), 243-256.
M. S. Mahadeva Naika and B. N. Dharmendra, On some new general theorems for the explicit evaluations of Ramanujan's remarkable product of theta-function Ramanujan J. 15(3) (2008), 349-366. https://doi.org/10.1007/s11139-007-9081-1
M. S. Mahadeva Naika, K. Sushan Bairy and S. Chandankumar, On Some Explicit evaluation of the ratios of Ramanujan's theta-function (Communicated).
M. S. Mahadeva Naika, B. N. Dharmendra and K. Shivashankara, On some new explicit evaluations of Ramanujan's remarkable product of theta-function, South East Asian J. Math. Math. Sci. 5(1) (2006), 107-119.
M. S. Mahadeva Naika and M. C. Maheshkumar, Explicit evaluations of Ramanujan's remarkable product of thetafunction, Adv. Stud. Contemp. Math., 13(2) (2006), 235-254.
M. S. Mahadeva Naika, M. C. Maheshkumar and K. Sushan Bairy, General formulas for explicit evaluations of Ramanujan's cubic continued fraction, Kyungpook Math. J., 49(3) (2009), 435-450. https://doi.org/10.5666/KMJ.2009.49.3.435
M. S. Mahadeva Naika, M. C. Maheshkumar and K. Sushan Bairy, On some remarkable product of theta-function, Aust. J. Math. Anal. Appl., 5(1) (2008), 1-15. https://doi.org/10.1007/s11139-007-9081-1
Nipen Saikia, Some Properties, Explicit Evaluation, and Applications of Ramanujan's Remarkable Product of ThetaFunctions, Acta Math Vietnam, Journal of Mathematics, DOI 10.1007/s40306-014-0106-8, (2015). https://doi.org/10.1007/s40306-014-0106-8
S. Y. Kang, Some theorems on the Rogers-Ramanujan continued fraction and associated theta function identities in Ramanujan's lost notebook. Ramanujan J., 3 (1) (1999), 91-11.
S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.
S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 1988.
Copyright (c) 2021 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
