Abstract and Applied Analysis

Dynamics of a continued fraction of Ramanujan with random coefficients

Jonathan M. Borwein and D. Russell Luke

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Abstract

We study a generalization of a continued fraction of Ramanujan with random, complex-valued coefficients. A study of the continued fraction is equivalent to an analysis of the convergence of certain stochastic difference equations and the stability of random dynamical systems. We determine the convergence properties of stochastic difference equations and so the divergence of their corresponding continued fractions.

Article information

Source
Abstr. Appl. Anal., Volume 2005, Number 5 (2005), 449-467.

Dates
First available in Project Euclid: 25 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1122298479

Digital Object Identifier
doi:10.1155/AAA.2005.449

Mathematical Reviews number (MathSciNet)
MR2201037

Zentralblatt MATH identifier
1134.11332

Citation

Borwein, Jonathan M.; Luke, D. Russell. Dynamics of a continued fraction of Ramanujan with random coefficients. Abstr. Appl. Anal. 2005 (2005), no. 5, 449--467. doi:10.1155/AAA.2005.449. https://projecteuclid.org/euclid.aaa/1122298479


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References

  • M. Abramowitz and I. A. Stegun, (eds.), Handbook of Mathematical Functions, 9th ed., Dover, New York, 1972.
  • D. Borwein, J. M. Borwein, R. Crandall, and R. Mayer, On the dynamics of certain recurrence relations, to appear in Ramanujan J.
  • J. M. Borwein and R. Crandall, On the Ramanujan AGM fraction, II: The complex-parameter case, Experiment. Math. 13 (2004), no. 3, 287--295.
  • J. M. Borwein, R. Crandall, and G. Fee, On the Ramanujan AGM fraction, I: The real-parameter case, Experiment. Math. 13 (2004), no. 3, 275--285.
  • J. M. Borwein and D. R. Luke, Dynamics of generalizations of the AGM continued fraction of Ramanujan. Part I: divergence, 2004, download: http://eprints.cecm.sfu.ca/archive/00000261.
  • G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, Clarendon Press, Oxford University Press, New York, 1992.
  • L. Lorentzen, A convergence question inspired by Stieltjes and by value sets in continued fraction theory, J. Comput. Appl. Math. 65 (1995), no. 1--3, 233--251.
  • L. Lorentzen and H. Waadeland, Continued Fractions with Applications, Studies in Computational Mathematics, vol. 3, North-Holland, Amsterdam, 1992.
  • E. M. E. Wermuth, Some elementary properties of infinite products, Amer. Math. Monthly 99 (1992), no. 6, 530--537.
  • A. Zygmund, Trigonometric series, 2nd ed., vols. I, II, Cambridge University Press, New York, 1959.