Linear fractional transformations and nonlinear leaping convergents of some continued fractions

Abstract

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.