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On evaluations of the cubic continued fraction by modular equations of degree 9

Jinhee Yi|Dae Hyun Paek
한국수학교육학회|2016.08|이론수학과 교직수학 (Theoretical Mathematics and Pedagogical Mathematics)|제23권 제3호|pp. 223-236

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참고문헌

키워드

continued fractionmodular equationstheta functions.
주제분류수학

초록

We show how to evaluate the cubic continued fraction $G(e^{-\pi\sqrt n})$ and $G(-e^{-\pi\sqrt n})$ for $n=4^m,\, 4^{-m},\,2\cdot4^m,\,$ and $2^{-1}\cdot4^{-m}$ for some nonnegative integer $m$ by using modular equations of degree 9. We then find some explicit values of them.

Abstract

We show how to evaluate the cubic continued fraction $G(e^{-\pi\sqrt n})$ and $G(-e^{-\pi\sqrt n})$ for $n=4^m,\, 4^{-m},\,2\cdot4^m,\,$ and $2^{-1}\cdot4^{-m}$ for some nonnegative integer $m$ by using modular equations of degree 9. We then find some explicit values of them.

참고문헌 (9)

  1. B.C. Berndt, "Number Theory in the Spirit of Ramanujan", American Mathematical Society (2006)
  2. B.C. Berndt, "Ramanujan's Notebooks, Part III" (1991), Springer-Verlag
  3. B.C. Berndt, "Ramanujan's class invariants and cubic con-tinued fraction", Acta Arith, 73, pp.67~85 (1995)
  4. H.H. Chan, "Ramanujan's cubic continued fraction", Acta Arith, 73, pp.343~355 (1995)
  5. 백대현, "On some modular equations and their applications II", 대한수학회보, 50(4), pp.1221~1233 (2013), 대한수학회
  6. J. Yi, "The Construction and Applications of Modular Equations", Ph. D, pp.2001
  7. J. Yi, "Theta-function identities and the explicit formulas for theta-function and their applications", J. Math. Anal. Appl, 292, pp.381~400 (2004)
  8. 이진희, "On Some Modular Equations and Their Applications I", 대한수학회보, 50(3), pp.761~776 (2013), 대한수학회
  9. J. Yi, "The explicit formulas and evaluations of Ramanujan's theta-function Ã", J. Math. Anal. Appl, 321, pp.157~181 (2006)

관련 논문AI추천

ON EVALUATIONS OF THE CUBIC CONTINUED FRACTION BY MODULAR EQUATIONS OF DEGREE 3Dae Hyun Paek, Jinhee Yi, Yong Jin Shin한국수학교육학회 · 이론수학과 교직수학2018 · 제25권 제1호On Evaluations of the Cubic Continued Fraction by Modular Equations of Degree 3 Revisited백대현, 안지원, 이강훈, 이진희한국수학교육학회 · 이론수학과 교직수학2024 · 제31권 제2호On some modular equations and their applications II백대현, 이진희대한수학회 · 대한수학회보2013 · 제50권 제4호On Some Modular Equations and Their Applications I김정환, 백대현, 유재명, 이성희, 이진희, 조만기대한수학회 · 대한수학회보2013 · 제50권 제3호Evaluations of the cubic continued fraction by some theta function identities백대현, 이진희강원경기수학회 · 한국수학논문집2019 · 제27권 제4호Some 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 제2호

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