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A counterexample to a continued fraction conjecture

Short, Ian (2006). A counterexample to a continued fraction conjecture. Proceedings of the Edinburgh Mathematical Society. Series II, 49(3) pp. 735–737.

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DOI (Digital Object Identifier) Link: http://doi.org/10.1017/S0013091505000581
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Abstract

It is known that if a ∈ ℂ \(−∞,−¼) and ana as n → ∞, then the infinite continued fraction with coefficients a1, a2,... converges. A conjecture has been recorded by Jacobsen et al., taken from the unorganized portions of Ramanujan’s notebooks, that if a ∈ (−∞,−¼) and an → a as n→∞, then the continued fraction diverges. Counterexamples to this conjecture for each value of a in (−∞,−¼) are provided. Such counterexamples have already been constructed by Glutsyuk, but the examples given here are significantly shorter and simpler.

Item Type: Journal Article
Copyright Holders: 2006 Edinburgh Mathematical Society
ISSN: 0013-0915
Keywords: Primary 40A15; 30B70; continued fractions; Möbius transformations; iteration; dynamics
Academic Unit/Department: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 22454
Depositing User: Ian Short
Date Deposited: 30 Jul 2010 14:03
Last Modified: 03 Aug 2016 16:19
URI: http://oro.open.ac.uk/id/eprint/22454
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