Abstract.
In this paper we first give the value of a periodic continued fraction which was recorded incorrectly by Ramanujan on page 341 of his lost notebook. Next, we describe several pairs of equivalent continued fractions in which one is the odd part of the other. One of the results is for the Rogers-Ramanujan continued fraction which was recently proved by Berndt and Yee. Finally, using the Bauer-Muir transformation we prove the equivalence of two continued fractions. One was recorded on page 44 in Ramanujan’s lost notebook, and the other is found in the unorganized pages at the end of Ramanujan’s second notebook.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G Andrews (1968) ArticleTitleOn q-difference equations for certain well-poised basic hypergeometric series. Quart J Math (Oxford) 19 433–447
Andrews G, Berndt B, Jacobsen L, Lamphere R (1992) The Continued Fractions Found in the Unorganized Portions of Ramanujan’s Notebooks. Memoir Amer Math Soc 99: No 477
G Andrews B Berndt J Sohn A Yee A Zaharescu (2003) ArticleTitleOn Ramanujan’s continued fraction for (q2;q3)∞/(q;q3)∞. Trans Amer Math Soc 355 2397–2411 Occurrence Handle10.1090/S0002-9947-02-03155-0
G Bauer (1872) ArticleTitleVon einem Kettenbruch Eulers und einem Theorem von Wallis. Abh Bayr Akad Wiss 11 96–116
Berndt B (1991) Ramanujan’s Notebooks, Part III. New York: Springer
Berndt B (1998) Ramanujan’s Notebooks, Part V. New York: Springer
Berndt B (2003) Personal communication to J. Sohn
B Berndt S Huang J Sohn S Son (2000) ArticleTitleSome theorems on the Rogers-Ramanujan continued fraction in Ramanujan’s lost notebook. Trans Amer Math Soc 352 2157–2177 Occurrence Handle10.1090/S0002-9947-00-02337-0
B Berndt A Yee (2003) ArticleTitleOn the generalized Roges-Ramanujan continued fraction. Ramanujan J 7 321–331 Occurrence Handle10.1023/A:1026215700375
Brillhart J (2002) Email to BC Berndt, January 27
L Jacobsen (1990) ArticleTitleOn the Bauer-Muir transformation for continued fractions and its applications. J Math Anal Appl 152 496–514 Occurrence Handle10.1016/0022-247X(90)90080-Y
Jones W, Thron W (1980) Continued Fractions: Analytic Theory and Applications. Reading: Addison-Wesley
Lorentzen L, Waadeland H (1992) Continued Fractions with Applications. Amsterdam: North Holland
Muir T (1877) A theorem in continuants. – Extension of a theorem in continuants, with an important application. London Edinburgh Dublin Philos Mag J Sci 5: 137 and 360
Ramanujan S (1957) Notebooks (2 volumes). Bombay: Tata Institute
Ramanujan S (1988) The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa
Author information
Authors and Affiliations
Additional information
This work was supported by Yonsei University Research Fund of 2003.
Rights and permissions
About this article
Cite this article
Lee, J., Sohn, J. Some Continued Fractions in Ramanujan’s Lost Notebook. Mh Math 146, 37–48 (2005). https://doi.org/10.1007/s00605-005-0304-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-005-0304-5