Dependence with complete connections and the Gauss-Kuzmin theorem for N-continued fractions

We consider a family $\{T_N:N\ge 1\}$ of interval maps as generalizations of the Gauss transformation. For the continued fraction expansion arising from $T_N$, we solve its Gauss-Kuzmin-type problem by applying the theory of random systems with complete connections by Iosifescu.