The Unreasonable Effectualness of Continued Function Expansions

2002/06/17 by Greg Martin, Martin, Greg
Mathematics · #11J70 (40A15) #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11J70

paper · pdf · doi:10.48550/arxiv.math/0206166

11 pages

arxiv created 2002/06/17 · arxiv updated 2009/11/30

Abstract

Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties.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 behavior.

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