论文标题
线性分数转换和一些持续分数的非线性跳跃收敛量
Linear fractional transformations and non-linear leaping convergents of some continued fractions
论文作者
论文摘要
对于$α_0= \ left [a_0，a_1，\ ldots \ right] $无限的持续分数和$σ$线性分数变换，我们研究了$σ（α_0）$的持续分数扩展及其收敛量。我们为四个持续分数的一般家庭提供$σ（α_0）$的持续分数扩展，当$ \ weft | \ detσ\ right |时= 2 $。我们还发现$σ（α_0）$的收敛之间的非线性复发关系，这使我们能够突出$α_0$和$σ（α_0）$之间的收敛之间的关系。最后，我们将结果应用于一些特殊且经过充分研究的持续分数，例如Hurwitzian和Tasoevian的分数，这是关于从非线性功能提供的步骤的第一个研究。
For $α_0 = \left[a_0, a_1, \ldots\right]$ an infinite continued fraction and $σ$ a linear fractional transformation, we study the continued fraction expansion of $σ(α_0)$ and its convergents. We provide the continued fraction expansion of $σ(α_0)$ for four general families of continued fractions and when $\left|\det σ\right| = 2$. We also find nonlinear recurrence relations among the convergents of $σ(α_0)$ which allow us to highlight relations between convergents of $α_0$ and $σ(α_0)$. 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.
