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New ramanujan’s remarkable product of Theta-function and their explicit evaluations

B. N. DHARMENDRA
장전수학회 (Jangjeon Mathematical Society)|2019.10|Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집) (Proceedings of the Jangjeon Mathematical Society)|제22권 제4호|pp. 517-528|DOI 10.17777/pjms.2019.22.4.002

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키워드

Continued fractionModular equationClass invariantThetafunction
주제분류수학

초록

In this article, we de ne dm;n for any positive real numbers m and n involving Ramanujan's product of theta-functions (q) and f(􀀀q2), 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 dm;n and its explicit values.

Abstract

In this article, we de ne dm;n for any positive real numbers m and n involving Ramanujan's product of theta-functions (q) and f(􀀀q2), 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 dm;n and its explicit values.

참고문헌 (15)

  1. B. C. Berndt, "Ramanujan's Notebooks, Part III" (1991), Springer-Verlag
  2. B. C. Berndt, "Ramanujan's Notebooks, Part IV" (1994), Springer-Verlag
  3. B. C. Berndt, "Ramanujan's Notebooks, Part V" (1997), Springer-Verlag
  4. B. C. Berndt, "Ramanujan's remarkable product of the theta-function", Proc. Edinburgh Math. Soc., 40, pp.583~612 (1997)
  5. M. S. Mahadeva Naika, "Some theorems on Ramanujan's cubic continued fraction and related identities", Tamsui Oxf. J. Math. Sci, 24(3), pp.243~256 (2008)
  6. M. S. Mahadeva Naika, "On some new general theorems for the explicit evaluations of Ramanujan's remarkable product of theta-function", Ramanujan J, 15(3), pp.349~366 (2008)
  7. M. S. Mahadeva Naika, "On some new explicit evaluations of Ramanujan's remarkable product of theta-function", South East Asian J. Math. Math. Sci, 5(1), pp.107~119 (2006)
  8. M. S. M. Naika, "Explicit evaluations of Ramanujan's remarkable product of theta-function", Advanced Studies in Contemporary Mathematics, 13(2), pp.235~254 (2006), 장전수학회
  9. M. S. Mahadeva Naika, "General Formulas for Explicit Evaluations of Ramanujan’s Cubic Continued Fraction", Kyungpook Mathematical Journal, 49(3), pp.435~450 (2009), 경북대학교 자연과학대학 수학과
  10. M. S. Mahadeva Naika, "On some remarkable product of theta-function", Aust. J. Math. Anal. Appl., 5(1), pp.1~15 (2008)
  11. M. S. Mahadeva Naika, "Some Modular Equations in The Form of Schlafli", Italian Journal of Pure and Applied Mathematics, 30, pp.233~252 (2013)
  12. Nipen Saikia, "Some Properties, Explicit Evaluations, and Applications of Ramanujan’s Remarkable Product of Theta Functions", Acta Mathematica Vietnamica, 41(1), pp.133~142 (2016), Springer Science and Business Media LLC[DOI]
  13. 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), pp.91~11 (1999)
  14. S. Ramanujan, "Notebooks (2 volumes)" (1957), Tata Institute of Fundamental Research
  15. S. Ramanujan, "lost notebook and other unpublished papers" (1988), Narosa

관련 논문AI추천

Some Result on Analogous to Ramanujan’s Remarkable Product of Theta FunctionB. N. DHARMENDRA, R. Raghavendra장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2020 · 제23권 제3호ON SOME NEW MODULAR RELATIONS FOR RAMANUJAN'S REMARKABLE PRODUCT OF THETA FUNCTIONSB. HEMANTHKUMAR, M. S. MAHADEVA NAIKA, S. Chandankumar장전수학회 · Advanced Studies in Contemporary Mathematics2013 · 제23권 제3호New identities for ratios of Ramanujan’s theta functionK. Sushan Bairy, M. S. MAHADEVA NAIKA, S. Chandankumar장전수학회 · Advanced Studies in Contemporary Mathematics2017 · 제27권 제1호Explicit evaluations of Ramanujan's remarkable product of theta-functionM. C. Maheshkumar, M. S. M. Naika장전수학회 · Advanced Studies in Contemporary Mathematics2006 · 제13권 제2호CERTAIN MODULAR RELATIONS FOR REMARKABLE PRODUCT OF THETA-FUNCTIONSK. Sushan Bairy, M. S. MAHADEVA NAIKA, N. P. SUMAN장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2014 · 제17권 제3호On some new modular equations of degree 9 and their applicationsM. Manjunatha, M. S. M. Naika, S. Chandankumar장전수학회 · Advanced Studies in Contemporary Mathematics2012 · 제22권 제1호

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