Commutator subgroups of the power subgroups of generalized Hecke groups

Let $p$, $q\geq 2$ be relatively prime integers and let $H_{p,q}$ be the generalized Hecke group associated to $p$ and $q$. The generalized Hecke group $H_{p,q}$ is generated by $X(z)=-(z-\lambda _{p})^{-1}$ and $Y(z)=-(z+\lambda_{q})^{-1}$ where $\lambda _{p}=2\cos \frac{\pi }{p}$ and $\lambda_{q}=2\cos \frac{\pi }{q}$. In this paper, for positive integer $m$, we study the commutator subgroups $(H_{p,q}^{m})'$ of the power subgroups $H_{p,q}^{m}$ of generalized Hecke groups $H_{p,q}$. We give an application related with the derived series for all triangle groups of the form $(0;p,q,n)$, for distinct primes $p$, $q$ and for positive integer $n$.