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The Ramanujan Journal

, Volume 16, Issue 3, pp 285–304 | Cite as

Dynamics of a Ramanujan-type continued fraction with cyclic coefficients

  • Jonathan M. Borwein
  • D. Russell LukeEmail author
Article
  • 47 Downloads

Abstract

We study several generalizations of the AGM continued fraction of Ramanujan inspired by a series of recent articles in which the validity of the AGM relation and the domain of convergence of the continued fraction were determined for certain complex parameters (Borwein et al., Exp. Math. 13, 275–286, 2004, Ramanujan J., in press, 2004; Borwein and Crandall, Exp. Math. 12, 287–296, 2004). A study of the AGM continued fraction is equivalent to an analysis of the convergence of certain difference equations and the stability of dynamical systems. Using the matrix analytical tools developed in 2004, we determine the convergence properties of deterministic difference equations and so divergence of their corresponding continued fractions.

Keywords

Continued fractions Ramanujan AGM relation Difference equations Matrix analysis 

Mathematics Subject Classification (2000)

11J70 40A15 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Faculty of Computer ScienceDalhousie UniversityHalifaxCanada
  2. 2.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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