Some modular relation on analogous of Ramanujan´´´'s remarkable product of theta-function
DOI:
https://doi.org/10.5269/bspm.50610Resumen
In this article, we derive new modular relations on Ramanujan's product of theta-functions $\phi(q)$ and $f(-q^2)$, which is analogous to Ramanujan's remarkable product of theta-functions and their explicit evaluations.
Referencias
1. B. C. Berndt, Ramanujan's Notebooks, Part III, Springer Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0965-2
2. B. C. Berndt, Ramanujan's Notebooks, Part IV, Springer-Verlag, New York, 1994. https://doi.org/10.1007/978-1-4612-0879-2
3. B. C. Berndt, Ramanujan's Notebooks, Part V, Springer-Verlag, New York, 1997.
4. 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
5. B. N. Dharmendra, New Ramanujan's Remarkable Product of Theta-Function and Their Explicit Evaluations. Proc. Jangjeon Math. Soc. 22(4) (2019), 517-528.
6. N. D. Baruha, Modular Equations for Ramanujan's Cubic Continued Fraction,Journal of Mathematical Analysis and Applications, 268 (2002), 244-255. https://doi.org/10.1006/jmaa.2001.7823
7. 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.
8. M. S. Mahadeva Naika, S. Chandankumar and K. Sushan Bairy, New Identites for Ratios of Ramanujan's theta-function, Advanced Studies in Contempary Mathematics, 27(1) (2017), 131-146.
9. M. S. Mahadeva Naika, K. Sushan Bairy and S. Chandankumar, On Some Explicit evaluation of the ratios of Ramanujan's theta-function, Bull. Allahabad Math. Soc. 29(1) (2014) 53-86.
10. 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
11. M. S. Mahadeva Naika, B. N. Dharmendra and S. Chandankumar, New Modular Relations for Ramanujan Parameter µ(q), IJPAM, 74(4) (2012), 413-435.
12. 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 .
13. 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.
14. 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
15. 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
16. Nipen Saikia, Some Properites, 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
17. 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.
18. S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.
19. S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 1988.
2. B. C. Berndt, Ramanujan's Notebooks, Part IV, Springer-Verlag, New York, 1994. https://doi.org/10.1007/978-1-4612-0879-2
3. B. C. Berndt, Ramanujan's Notebooks, Part V, Springer-Verlag, New York, 1997.
4. 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
5. B. N. Dharmendra, New Ramanujan's Remarkable Product of Theta-Function and Their Explicit Evaluations. Proc. Jangjeon Math. Soc. 22(4) (2019), 517-528.
6. N. D. Baruha, Modular Equations for Ramanujan's Cubic Continued Fraction,Journal of Mathematical Analysis and Applications, 268 (2002), 244-255. https://doi.org/10.1006/jmaa.2001.7823
7. 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.
8. M. S. Mahadeva Naika, S. Chandankumar and K. Sushan Bairy, New Identites for Ratios of Ramanujan's theta-function, Advanced Studies in Contempary Mathematics, 27(1) (2017), 131-146.
9. M. S. Mahadeva Naika, K. Sushan Bairy and S. Chandankumar, On Some Explicit evaluation of the ratios of Ramanujan's theta-function, Bull. Allahabad Math. Soc. 29(1) (2014) 53-86.
10. 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
11. M. S. Mahadeva Naika, B. N. Dharmendra and S. Chandankumar, New Modular Relations for Ramanujan Parameter µ(q), IJPAM, 74(4) (2012), 413-435.
12. 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 .
13. 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.
14. 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
15. 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
16. Nipen Saikia, Some Properites, 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
17. 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.
18. S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.
19. S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 1988.
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2022-02-07
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