PURELY PERIODIC ROSEN CONTINUED FRACTIONS

Abstract

We consider the two Hecke groups $G_4$ and $G_6$ generated by the transformations $S$ and $T$ defined by $S(z)=z+{\lambda }_m$ and $T(z)=-1∕z$ where ${\lambda }_m=2\text{ cos}(\pi ∕m)$ with $m\in \{4,6\}$. We give a full characterization of purely periodic Rosen continued fractions over $G_4$ and $G_6$. Finally, we end by finding a family of examples of purely periodic Rosen expansions of period length two and some related examples.