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The unreasonable effectualness of continued function expansions

Part of: Diophantine approximation, transcendental number theory Convergence and divergence of infinite limiting processes

Published online by Cambridge University Press:  09 April 2009

Greg Martin
Greg Martin
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada e-mail: [email protected]
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Abstract

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Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties. For instance, we might ask that algebraic numbers of a given degree have periodic expansions, just as quadratic irrationals have periodic continued fractions; or we might ask that familiar transcendental constants such as e or π have periodic or terminating expansions. In this paper, we show that there exist such generalized continued function expansions with essentially any desired behaviour.

MSC classification

Secondary: 11J70: Continued fractions and generalizations 40A15: Convergence and divergence of continued fractions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 3 , December 2004 , pp. 305 - 320
Copyright
Copyright © Australian Mathematical Society 2004

References

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