A dependence with complete connections approach to generalized R\'enyi continued fractions

We introduce and study in detail a special class of backward continued fractions that represents a generalization of R\'enyi continued fractions. We investigate the main metrical properties of the digits occurring in these expansions and we construct the natural extension for the transformation that generates the R\'enyi-type expansion. Also we define the associated random system with complete connections whose ergodic behavior allows us to prove a variant of Gauss-Kuzmin-type theorem.