논문KCI 등재
Relations Between Ramanujan's Cubic Continued Fraction and a Continued Fraction of Order 12 and its Evaluations
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
주제분류수학
초록
In the present paper, we establish relationship between continued fraction U(-q) of order 12 and Ramanujan's cubic continued fraction G(-q) and G(qn) for n = 1; 2; 3; 5 and 7. Also we evaluate U(q) and U(-q) by using two parameters for Ramanujan's theta-functions and their explicit values.
Abstract
In the present paper, we establish relationship between continued fraction U(-q) of order 12 and Ramanujan's cubic continued fraction G(-q) and G(qn) for n = 1; 2; 3; 5 and 7. Also we evaluate U(q) and U(-q) by using two parameters for Ramanujan's theta-functions and their explicit values.
참고문헌 (31)
- C. Adiga, "Some new modular relations for the Rogers-Ramanujan type functions of order eleven with applications to partitions", J. of Number Theory, 158, pp.281~297 (2016)
- C. Adiga, "A Continued fraction of Ramanujan and some Ramanujan-Weber class invariants", Filomat, 31(13), pp.3975~3997 (2017)
- N. D. Baruah, "Some general theorems on the explicit evaluations of Ramanujan's cubic continued fraction", J. Comput. Appl. Math, 160, pp.37~51 (2003)
- N. D. Baruah, "Two parameters for Ramanujan's theta-functions and their explicit values", Rocky Mountain J. Math, 37(6), pp.1747~1790 (2007)
- B. C. Berndt, "Ramanujan's Notebooks. Part III" (1991), Springer-Verlag
- B. C. Berndt, "Ramanujan's forty identities for the Rogers-Ramanujan functions", Mem. Amer. Math. Soc, 188(880), pp.1~96 (2007)
- S. Bhargava, "Evaluations of Ramanujan-Weber class invariant <TEX>$g_{n}^{\ast}$<", TEX>, 2005, pp.72(1-4), 115~127
- A. J. F. Biagioli, "A proof of some identities of Ramanujan using modular forms", Glasgow Math. J, 31, pp.271~295 (1989)
- B. J. Birch, "A look back at Ramanujan's notebooks", Math. Proc. Cambridge Philos. Soc, 78, pp.73~79 (1975)
- D. Bressoud, "Proof and generalization of certain identities conjectured by Ramanujan", Ph.D, pp.1977
- H. H. Chan, "On Ramanujan's cubic continued fraction", Acta Arith, 73, pp.343~345 (1995)
- H. H. Chan, "On the Ramanujan-Gollnitz-Gordon continued fraction", Ramanujan J, 1, pp.75~90 (1997)
- B. Cho, "Arithmetic of the Ramanujan-Gollnitz-Gordon continued fraction", J. Number Theory, 129(4), pp.922~947 (2009)
- H. Gollnitz, "Partitionen mit Differenzenbendingungen", J. Reine Angew. Math, 225, pp.154~190 (1967)
- B. Gordon, "Some continued fractions of the Rogers-Ramanujan type", Duke Math. J, 32, pp.741~748 (1965)
- G. H. Hardy, "Ramanujan: twelve lectures on subjects suggested by his life and work" (1959), Chelsea Publishing Company
- M. S. M. Naika, "On some new explicit evaluations of class invariants", Vietnam J. Math, 36(1), pp.103~124 (2008)
- M. S. M. Naika, "A continued fraction of order twelve", Cent. Eur. J. Math, 6(3), pp.393~404 (2008)
- S. Ramanujan, "Algebraic relations between certain infinite products", Proc. London Math. Soc, 2, pp.18~ (1920)
- S. Ramanujan, "Collected papers of Srinivasa Ramanujan" (2000), American Mathematical Society
- S. Ramanujan, "Notebooks (2 Volumes)" (1957), Tata Institute of Fundamental Research
- S. Ramanujan, "The Lost notebook and other unpublished papers" (1988), Narosa Publishing House
- L. J. Rogers, "On a type of modular relation", Proc. London Math. Soc, 19, pp.387~397 (1920)
- H. M. Srivastava, "Some Relationships Between q-Product Identities, Combinatorial Partition Identities and Continued-Fraction Identities", Advanced Studies in Contemporary Mathematics, 25(3), pp.265~272 (2015), 장전수학회
- H. M. Srivastava, "Some theta function identities related to Jacobi's triple product identity", Eur. J. Pure Appl. Math, 11, pp.1~9 (2018)[DOI]
- K. R. Vasuki, "On a continued fraction of order 12", Ukrainian Math. J, 62(12), pp.1866~1878 (2011)
- K. R. Vasuki, "Certain identities for Ramanujan-Gollnitz-Gordon continued fraction", J. Comput. App. Math, 187, pp.87~95 (2006)
- K. R. Vasuki, "A new identity for the Rogers-Ramanujan continued fraction", J. Appl. Math. Anal. Appl, 2(1), pp.71~83 (2006)
- G. N. Watson, "Proof of certain identities in combinatory analysis", J. Indian Math. Soc, 20, pp.57~69 (1933)
- H. Weber, "Lehrbuch der Algebra" (1961), Chelsea
- J. Yi, "Theta function identities and the explicit formulas for theta functions and their applications", J. Math. Anal. Appl, 292(2), pp.381~400 (2004)
관련 논문AI추천
Some 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 제2호Evaluations of the cubic continued fraction by some theta function identities백대현, 이진희강원경기수학회 · 한국수학논문집2019 · 제27권 제4호Evaluations of the cubic continued fraction by some theta function identities: Revisited백대현한국수학교육학회 · 이론수학과 교직수학2021 · 제28권 제1호General Formulas for Explicit Evaluations of Ramanujan’s Cubic Continued FractionK. Sushan Bairy, M. C. Maheshkumar, M. S. Mahadeva Naika경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2009 · 제49권 제3호Evaluations of the Rogers-Ramanujan continued Fraction by Theta-function Identities Revisited백대현, Jinhee Yi한국수학교육학회 · 이론수학과 교직수학2022 · 제29권 제3호Certain identities for a continued fraction of RamanujanK. S. Bairy, M. S. M. Naika, S. Chandankumar장전수학회 · Advanced Studies in Contemporary Mathematics2014 · 제24권 제1호
이 간행물의 다른 논문 (14건)
Extensions of Strongly α-semicommutative RingsLiu Zhongkui, Ayoub Elshokry, Eltiyeb Alipp. 203-219Bounds for the First Zagreb Eccentricity Index and First Zagreb Degree Eccentricity IndexVeena Mathad, Padmapriya Ppp. 221-229Sharp Coefficient Bounds for the Quotient of Analytic Functions박지향, 조낙은, Virendra Kumarpp. 231-242Certain Subclasses of k−uniformly Functions Involving the Generalized Fractional Differintegral OperatorTamer Mohamed Seoudypp. 243-255First Order Differential Subordinations for Carathéodory FunctionsShweta Gandhi, SUSHIL KUMAR, V. Ravichandranpp. 257-270
