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

For $\alpha_0 = \left[a_0, a_1, \ldots\right]$ 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 $\left|\det \sigma\right| = 2$. 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.