Generalized Continued Logarithms and Related Continued Fractions
Jonathan M. Borwein
University of Newcastle
Callaghan NSW 2308
Kevin G. Hare and Jason G. Lynch
Department of Pure Mathematics
University of Waterloo
200 University Ave. W.
Waterloo, ON N2L 3G1
We study continued logarithms, as introduced by Gosper and studied by
Borwein et al. After providing an overview of the type I and type II
generalizations of binary continued logarithms introduced by Borwein et
al., we focus on a new generalization to an arbitrary integer base b.
We show that all of our so-called type III continued logarithms
converge and all rational numbers have finite type III continued
logarithms. As with simple continued fractions, we show that the
continued logarithm terms, for almost every real number, follow a
specific distribution. We also generalize Khinchin's constant from
simple continued fractions to continued logarithms, and show that these
logarithmic Khinchin constants have an elementary closed form. Finally,
we show that simple continued fractions are the limiting case of our
continued logarithms, and briefly consider how we could generalize
beyond continued logarithms.
Full version: pdf,
Received October 3 2016; revised versions received May 8 2017; May 9 2017.
Published in Journal of Integer Sequences, May 21 2017.
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