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.
On decimal and continued fraction expansions of a real number