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

For α0 = [a0, a1, ...] an infinite continued fraction and σ a linear fractional transformation, we study the continued fraction expansion of σ(α0) and its convergents. We provide the continued fraction expansion of σ(α0) for four general families of continued fractions and when |det σ| = 2. We also find nonlinear recurrence relations among the convergents of σ(α0) which allow us to highlight relations between convergents of α0 and σ(α0). 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.