한국수학교육학회지시리즈B:순수및응용수학 (The Pure and Applied Mathematics)

  • 제28권4호
  • /
  • Pages.377-386
  • /
  • 2021
  • /
  • 1226-0657(pISSN)
  • /
  • 2287-6081(eISSN)
한국수학교육학회 (Korean Society of Mathematical Education)
권호 표지
DOI QR코드

DOI QR Code

EVALUATIONS OF THE ROGERS-RAMANUJAN CONTINUED FRACTION BY THETA-FUNCTION IDENTITIES

  • Paek, Dae Hyun (Department of Mathematics Education, Busan National University of Education)
  • 투고 : 2021.07.23
  • 심사 : 2021.09.01
  • 발행 : 2021.11.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we use theta-function identities involving parameters 𝑙5,n, 𝑙'5,n, and 𝑙'5,4n to evaluate the Rogers-Ramanujan continued fractions $R(e^{-2{\pi}{\sqrt{n/20}}})$ and $S(e^{-{\pi}{\sqrt{n/5}}})$ for some positive rational numbers n.

키워드

과제정보

This work is supported by a research grant from the Busan National University of Education in 2021.

참고문헌

  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: Ramanujan's Notebooks, Part IV. Springer-Verlag, New York, 1994.
  4. B.C. Berndt & H.H. Chan: Some values for the Rogers-Ramanujan continued fraction. Can. J. Math. 47 (1995), no. 5, 897-914. https://doi.org/10.4153/CJM-1995-046-5
  5. B.C. Berndt, H.H. Chan & L.C Zhang: Explicit evaluations of the Rogers-Ramanujan continued fraction. J. Reine Angew. Math. 480 (1996), 141-159.
  6. 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
  7. D.H. Paek & J. Yi: On some modular equations of degree 5 and their applications. Bull. Korean Math. Soc. 50 (2013), no. 4, 1315-1328. https://doi.org/10.4134/BKMS.2013.50.4.1315
  8. K.G. Ramanathan: On Ramanujan's continued fraction. Acta Arith. 43 (1984), 209-226. https://doi.org/10.4064/aa-43-3-209-226
  9. K.G. Ramanathan: On the Rogers-Ramanujan continued fraction. Proc. Indian Acad. Sci. Math. Sci. 93 (1984), 67-77. https://doi.org/10.1007/BF02840651
  10. K.G. Ramanathan: Ramanujan's continued fraction. Indian J. Pure Appl. Math. 16 (1985), 695-724.
  11. K.G. Ramanathan: Some applications of Kronecker's limit formula. J. Indian Math. Soc. 52 (1987), 71-89.
  12. N. Saikia: Some new general theorems for the explicit evaluations of the Rogers-Ramanujan continued fraction. Comput. Methods Funct. Theory 13 (2013), 597-611. https://doi.org/10.1007/s40315-013-0040-0
  13. J. Yi: Evaluations of the Rogers-Ramanujan continued fraction R(q) by modular equations. Acta Arith. 97 (2001), 103-127. https://doi.org/10.4064/aa97-2-2
  14. 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. https://doi.org/10.1016/j.jmaa.2005.07.062