Linear fractional transformations and non-linear leaping convergents of some continued fractions

For ${\alpha }_0=[a_0,a_1,\dots ]$ an infinite continued fraction and $\sigma$ a linear fractional transformation, we study the continued fraction expansion of $\sigma ({\alpha }_0)$ and its convergents. We provide the continued fraction expansion of $\sigma ({\alpha }_0)$ for four general families of continued fractions and when $\mid \det \sigma \mid =2$. We also find nonlinear recurrence relations among the convergents of $\sigma ({\alpha }_0)$ which allow us to highlight relations between convergents of ${\alpha }_0$ and $\sigma ({\alpha }_0)$. Finally, we apply our results to some special and well-studied continued fractions, like Hurwitzian and Tasoevian ones, giving a first study about leaping convergents having steps provided by nonlinear functions.