Generalized continued fraction expansions with constant partial denominators
Törmä, Topi (2018-12-21)
Törmä, Topi (2019) Generalized continued fraction expansions with constant partial denominators. Journal of the Australian Mathematical Society, 107(2), 272-288. https://doi.org/10.1017/S1446788718000332
This article has been published in a revised form in Journal of the Australian Mathematical Society, https://doi.org/10.1017/S1446788718000332. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © 2018 Australian Mathematical Publishing Association Inc.
https://rightsstatements.org/vocab/InC/1.0/
https://urn.fi/URN:NBN:fi-fe201903138735
Tiivistelmä
Abstract
We study generalized continued fraction expansions of the form \[\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}\] where \(N\) is a fixed positive integer and the partial numerators \(a_{i}\) are positive integers for all \(i\). We call these expansions \(\operatorname{dn}_{N}\) expansions and show that every positive real number has infinitely many \(\operatorname{dn}_{N}\) expansions for each \(N\). In particular, we study the \(\operatorname{dn}_{N}\) expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each \(N\), a \(\operatorname{dn}_{N}\) expansion with bounded partial numerators.
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