The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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ON EVALUATIONS OF THE CUBIC CONTINUED FRACTION BY MODULAR EQUATIONS OF DEGREE 3 REVISITED

  • Jinhee Yi (Department of Mathematics and Computer Science, Korea Science Academy of KAIST) ;
  • Ji Won Ahn (Korea Science Academy of KAIST) ;
  • Gang Hun Lee (Korea Science Academy of KAIST) ;
  • Dae Hyun Paek (Department of Mathematics Education, Busan National University of Education)
  • Received : 2023.10.11
  • Accepted : 2023.12.23
  • Published : 2024.05.31
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Abstract

We derive modular equations of degree 3 to find corresponding theta-function identities. We use them to find some new evaluations of $G(e^{-{\pi}{\sqrt{n}}})$ and $G(-e^{-{\pi}{\sqrt{n}}})$ for $n\,=\,\frac{25}{3{\cdot}4^{m-1}}$ and $\frac{4^{1-m}}{3{\cdot}25}$, where m = 0, 1, 2.

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References

  1. C. Adiga, T. Kim, M. S. Mahadeva Naika & H. S. Madhusudhan: On Ramanujan's cubic continued fraction and explicit evaluations of theta-functions. Indian J. pure appl. Math. 35 (2004), no. 9, 1047-1062. https://doi.org/10.48550/arXiv.math/0502323 
  2. C. Adiga, K. R. Vasuki & M. S. Mahadeva Naika: Some new explicit evaluations of Ramanujan's cubic continued fraction. New Zealand J. Math. 31 (2002), 109-114. 
  3. G. E. Andrews & B. C. Berndt: Ramanujan's lost notebooks, Part I. Springer, 2000. 
  4. N. D. Baruah: Modular equations for Ramanujan's cubic continued fraction. J. Math. Anal. Appl. 268 (2002), 244-255. doi:10.1006/jmaa.2001.7823 
  5. B. C. Berndt: Ramanujan's notebooks, Part III. Springer-Verlag, New York, 1991. 
  6. B. C. Berndt, H. H. Chan & L.-C. Zhang: Ramanujan's class invariants and cubic continued fraction. Acta Arith. 73 (1995), 67-85. DOI:10.4064/aa-73-1-67-85 
  7. H. H. Chan: On Ramanujan's cubic continued fraction. Acta Arith. 73 (1995), 343-355. DOI:10.4064/aa-73-4-343-355 
  8. D. H. Paek: Evaluations of the cubic continued fraction by some theta function identities: Revisited. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 28 (2021), no. 1, 27-42. http://dx.doi.org/10.7468/jksmeb.2021.28.1.27 
  9. D. H. Paek & J. Yi: On some modular equations and their applications II. Bull. Korean Math. Soc. 50 (2013), no. 4, 1211-1233. http://dx.doi.org/10.4134/BKMS.2013.50.4.1221 
  10. 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. http://dx.doi.org/10.7468/jksmeb.2016.23.3.223 
  11. D. H. Paek, Y. J. Shin & J. Yi: On evaluations of the cubic continued fraction by modular equations of degree 3. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 25 (2018), no. 1, 17-29. http://dx.doi.org/10.7468/jksmeb.2018.25.1.17 
  12. K. G. Ramanathan: On Ramanujan's continued fraction. Acta Arith. 43 (1984), 209-226. DOI:10.4064/aa-43-3-209-226 
  13. J. Yi: The construction and applications of modular equations. Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2001. 
  14. 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. http://dx.doi.org/10.4134/BKMS.2013.50.3.761 
  15. J. Yi, Y. Lee & D. H. Paek: The explicit formulas and evaluations of Ramanujan's theta-function ψ. J. Math. Anal. Appl. 321 (2006), 157-181. doi:10.1016/j.jmaa.2005.07.062 
  16. J. Yi & D. H. Paek: Evaluations of the cubic continued fraction by some theta function identities. Korean J. Math. 27 (2019), no. 4, 1043-1059. https://doi.org/10.11568/kjm.2019.27.4.1043