Linear fractional transformations and nonlinear 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.