GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

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.