The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

ON EVALUATIONS OF THE CUBIC CONTINUED FRACTION BY MODULAR EQUATIONS OF DEGREE 3

  • Paek, Dae Hyun (Department of Mathematics Education, Busan National University of Education) ;
  • Shin, Yong Jin (Department of Mechanical and Aerospace Engineering, Seoul National University) ;
  • Yi, Jinhee (Department of Mathematics and Computer Science, Korea Science Academy of KAIST)
  • Received : 2017.11.18
  • Accepted : 2018.02.23
  • Published : 2018.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

We find modular equations of degree 3 to evaluate some new values of the cubic continued fraction $G(e^{-{\pi}\sqrt{n}})$ and $G(-e^{-{\pi}\sqrt{n}})$ for $n={\frac{2{\cdot}4^m}{3}}$, ${\frac{1}{3{\cdot}4^m}}$, and ${\frac{2}{3{\cdot}4^m}}$, where m = 1, 2, 3, or 4.

Keywords

Acknowledgement

Supported by : Busan National University of Education

References

  1. B.C. Berndt: Number theory in the spirit of Ramanujan. American Mathematical Society, 2006.
  2. B.C. Berndt: Ramanujan's notebooks, Part III. Springer-Verlag, New York, 1991.
  3. B.C. Berndt, H.H. Chan & L.-C. Zhang: Ramanujan's class invariants and cubic continued fraction. Acta Arith. 73 (1995), 67-85. https://doi.org/10.4064/aa-73-1-67-85
  4. H.H. Chan: On Ramanujan's cubic continued fraction. Acta Arith. 73 (1995), 343-355. https://doi.org/10.4064/aa-73-4-343-355
  5. D.H. Paek & J. Yi: On some modular equations and their applications II. Bull. Korean Math. Soc. 50 (2013), no. 4, 1211-1233.
  6. D.H. Paek & J. Yi: On evaluations of the cubic continued fraction by modular equations of degree 9. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 23 (2016), no. 3, 223-236.
  7. J. Yi: The construction and applications of modular equations. Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2001.
  8. J. Yi, M.G. Cho, J.H. Kim, S.H. Lee, J.M. Yu & D.H. Paek: On some modular equations and their applications I. Bull. Korean Math. Soc. 50 (2013), no. 3, 761-776. https://doi.org/10.4134/BKMS.2013.50.3.761
  9. J. Yi, Y. Lee & D.H. Paek: The explicit formulas and evaluations of Ramanujan's theta-function. J. Math. Anal. Appl. 321 (2006), 157-181. https://doi.org/10.1016/j.jmaa.2005.07.062