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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.

This paper deals with a class of discrete-time Markov chains for which the invariant measures can be expressed in terms of generalized continued fractions. The representation covers a wide class of stochastic models and is well suited for numerical applications. The results obtained can easily be extended to continuous-time Markov chains.

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GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

Markov chains and generalized continued fractions

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GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

Markov chains and generalized continued fractions