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

### Generalized continued fraction expansions with constant partial denominators

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Author: Törmä, Topi1
Organizations: 1Research Unit of Mathematical Sciences, P.O. Box 8000, 90014 University of Oulu, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.3 MB)
Language: English
Published: Cambridge University Press, 2019
Publish Date: 2019-06-21
Description:

# 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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Series: Journal of the Australian Mathematical Society
ISSN: 1446-7887
ISSN-E: 1446-8107
ISSN-L: 1446-7887
Volume: 107
Issue: 2
Pages: 272 - 288
DOI: 10.1017/S1446788718000332