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Special Case on Rogers-Ramanujan Type Continued Fraction Identity

B. N. DHARMENDRA
장전수학회 (Jangjeon Mathematical Society)|2020.07|Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집) (Proceedings of the Jangjeon Mathematical Society)|제23권 제3호|pp. 331-340|DOI 10.17777/pjms.2020.23.3.005

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키워드

Modular equationsContinued fractionExplicit valuesRamanujans class invariants
주제분류수학

초록

In this article, we derive a new continued fraction d(q) by using a general continued fraction in Ramanujan's lost notebook. By using this continued fraction d(q), we establish modular relation between d(q) and d(qn), where n = 2, 3, 4, 5, 7, 11. We also establish some explicit values of d(q) by using Ramanujan's class invariant.

Abstract

In this article, we derive a new continued fraction d(q) by using a general continued fraction in Ramanujan's lost notebook. By using this continued fraction d(q), we establish modular relation between d(q) and d(qn), where n = 2, 3, 4, 5, 7, 11. We also establish some explicit values of d(q) by using Ramanujan's class invariant.

참고문헌 (13)

  1. C. Adiga, "On Some Rogers-Ramanujan Type Continued Fraction Identities", Math. Balkanica (N.S.), 12, pp.37~45 (1998)
  2. G. E. Andrews, "Ramanujan's Lost Notebook, Part I" (2005)
  3. B. C. Berndt, "Ramanujan's Notebooks, Part III" (1991), Springer-Verlag
  4. B. C. Berndt, "Ramanujan's Notebooks, Part IV" (1994), Springer-Verlag
  5. B. C. Berndt, "Ramanujan's Notebooks, Part V" (1998), Springer-Verlag
  6. W. N. Balley, "A note on certain q-identities", Q. J. Math, 12, pp.173~175 (1941)
  7. J. A. Daum, "The basic analog of Kummer's theorem", Bull. Amer. Math. Soc, 48, pp.711~713 (1942)
  8. M. S. Mahadeva Naika, "On Some New Explicit Evaluations of Class Invariants", Vietnam J. Math, 36(1), pp.103~124 (2018)
  9. M. S. Mahadeva Naika, "Some Modular Equations in The Form of Schlai", Ital. J. Pure Appl. Math, 30, pp.233~252 (2013)
  10. Nipen Saika, "Some result on a special case of a general continued fraction of Ramanujan", Ann. Univ. Ferrara, Sez. VII, Sci. Mat, 64(1), pp.165~183 (2018)
  11. S. Ramanujan, "Notebooks (2 volumes)" (1957), Tata Institute of Fundamental Research
  12. The `lost' notebook and other unpublished papers, "S. Ramanujan", ~ (1988)
  13. S. Ramanujan, "Modular equations and approximations to π", Q. J. Math., 45, pp.350~372 (1914)

관련 논문AI추천

Certain identities for a continued fraction of RamanujanK. S. Bairy, M. S. M. Naika, S. Chandankumar장전수학회 · Advanced Studies in Contemporary Mathematics2014 · 제24권 제1호SOME MODULAR RELATIONS BETWEEN RAMANUJAN’S FUNCTION V(q) AND V(qn)B. N. DHARMENDRA, M. C. MAHESH KUMAR장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2020 · 제23권 제3호Evaluations of the Rogers-Ramanujan continued Fraction by Theta-function Identities Revisited백대현, Jinhee Yi한국수학교육학회 · 이론수학과 교직수학2022 · 제29권 제3호Identities for the Rogers-Ramanujan continued fractionNayandeep Deka Baruah, Pranjal Talukdar대한수학회 · 대한수학회지2025 · 제62권 제1호Some more relations on Ramanujan's cubic continued fraction and a continued fraction of order 12B. R. S. Kumar, H. C. Vidya장전수학회 · Advanced Studies in Contemporary Mathematics2014 · 제24권 제2호Note on Modular relations for the Roger-Ramanujan type identities and representations for Jacobian identity최준상, M.P.Chaudhary영남수학회 · East Asian Mathematical Journal2015 · 제31권 제5호

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