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Generalized Continued Logarithms and Related Continued Fractions

Borwein, Jonathan M. and Hare, Kevin G. and Lynch, Jason G. (2016) Generalized Continued Logarithms and Related Continued Fractions. (Submitted)

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    Abstract

    We study continued logarithms as introduced by Bill Gosper and studied by J.Borwein et. al.. After providing an overview of the type I and type II generalizationsof binary continued logarithms introduced by Borwein et. al., we focus on a newgeneralization to an arbitrary integer baseb. We show that all of our so-called typeIII continued logarithms converge and all rational numbers have nite type III con-tinued logarithms. As with simple continued fractions, we show that the continuedlogarithm terms, for almost every real number, follow a speci c distribution. Wealso generalize Khinchine's constant from simple continued fractions to continuedlogarithms, and show that these logarithmic Khinchine constants have an elemen-tary closed form. Finally, we show that simple continued fractions are the limitingcase of our continued logarithms, and brie y consider how we could generalize pastcontinued logarithms.

    Item Type: Article
    Subjects: UNSPECIFIED
    Faculty: UNSPECIFIED
    Depositing User: Mrs Naghmana Tehseen
    Date Deposited: 05 Sep 2016 20:28
    Last Modified: 05 Sep 2016 20:28
    URI: https://docserver.carma.newcastle.edu.au/id/eprint/1739

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