Please use this identifier to cite or link to this item: http://hdl.handle.net/1959.13/928084
- Title
- On the dynamics of certain recurrence relations
- Author/Creator
-
Borwein, D.;
Borwein, J.;
Crandall, R.;
Mayer, R.
- Description
- In recent analyses the remarkable AGM continued fraction of Ramanujan—denoted R₁(a, b)—was proven to converge for almost all complex parameter pairs (a, b). It was conjectured that R₁ diverges if and only if (0 ≠ a = be iφ with cos² φ ≠ 1) or (a² = b² ∊ (−∞, 0)). In the present treatment we resolve this conjecture to the positive, thus establishing the precise convergence domain for R₁. This is accomplished by analyzing, using various special functions, the dynamics of sequences such as (tn) satisfying a recurrence tn = (tn₋₋₁ + (n - 1)κn₋₋₁ᵗtn₋₂)/n, where κn := a², b² 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 R1 , 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.
- Relation
- The Ramanujan Journal Vol. 13, Issue 1-3, p. 63-101
- Publisher Link
- http://dx.doi.org/10.1007/s11139-006-0243-3
- Date
- 2007
- Publisher
- Springer
- Keyword(s)
-
complex continued fractions;
dynamical systems;
arithmetic-geometric mean;
matrix analysis;
stability theory
- Resource Type
- journal article
- Identifier
- http://hdl.handle.net/1959.13/928084
- Identifier
- ISSN:1382-4090
- Language
- eng
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