The Ramanujan Journal

, Volume 13, Issue 1–3, pp 63–101 | Cite as

On the dynamics of certain recurrence relations

  • D. BorweinEmail author
  • J. Borwein
  • R. Crandall
  • R. Mayer


In recent analyses [3, 4] the remarkable AGM continued fraction of Ramanujan—denoted \({\cal R}_1\) (a, b)—was proven to converge for almost all complex parameter pairs (a, b). It was conjectured that \({\cal R}_1\) diverges if and only if (0≠ a = be i φ with cos 2φ ≠ 1) or (a 2 = b 2∊ (−∞, 0)). In the present treatment we resolve this conjecture to the positive, thus establishing the precise convergence domain for \({\cal R}_1\). This is accomplished by analyzing, using various special functions, the dynamics of sequences such as (t n ) satisfying a recurrence

$$ t_n = (t_{n-1} + (n-1) \kappa_{n-1}t_{n-2})/n, $$

where κ n a 2, b 2 as n be even, odd respectively.

As a byproduct, we are able to give, in some cases, exact expressions for the n-th convergent to the fraction \({\cal R}_1\), thus establishing some precise convergence rates. It is of interest that this final resolution of convergence depends on rather intricate theorems for complex-matrix products, which theorems evidently being extensible to more general continued fractions.


Complex continued fractions Dynamical systems Arithmetic-geometric mean Matrix analysis Stability theory 


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© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Western OntarioLondon OntarioCanada
  2. 2.Faculty of Computer Science Dalhousie UniversityHalifaxCanada
  3. 3.Center for Advanced ComputationReed CollegePortland
  4. 4.Department of MathematicsReed CollegePortland

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