Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T14:23:54.706Z Has data issue: false hasContentIssue false
  • English
  • Français

A continued fraction of Ramanujan

Part of: Series expansions Convergence and divergence of infinite limiting processes

Published online by Cambridge University Press:  09 April 2009

M. D. Hirschhorn
M. D. Hirschhorn
Affiliation:
School of Mathematics University of New South WalesKensington, NSW 2033, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a manuscript discovered in 1976 by George E. Andrews, Ramanujan states a formula for a certain continued fraction, without proof. In this paper we establish formulae for the convergents to the continued fraction, from which Ramanujan's result is easily deduced.

MSC classification

Secondary: 30B70: Continued fractions 40A15: Convergence and divergence of continued fractions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 29 , Issue 1 , February 1980 , pp. 80 - 86
Copyright
Copyright © Australian Mathematical Society 1980

References

Andrews, G. E. (1979), ‘An introduction to Ramanujan's “lost” notebook’, Amer. Math. Monthly 86, 89–108.Google Scholar
Eisenstein, G. (1844), ‘Transformations remarquables de quelques series’, Crelle's Journal 27, 193–197.Google Scholar
Gordon, B. (1965), ‘Some continued fractions of the Rogers–Ramanujan type’, Duke Math. J. 32, 741–748.Google Scholar
Ramanujan, S. (1919), ‘Proof of certain identities in combinatory analysis’, Proc. Camb. Phil. Soc. 19, 214–216.Google Scholar
Ramanujan, S. (1920?), unpublished manuscript.Google Scholar
Rogers, L. J. (1894), ‘Second memoir on the expansion of certain infinite products’, Proc. London Math. Soc..25, 318–343.Google Scholar
Watson, G. N., ‘A new proof of the Rogers–Ramanujan identities’, J. London Math. Soc. 4, 4–9.Google Scholar