论文标题
坚强的恩格尔系列和吕伦斯系列的持续分数带有标志
Continued fractions for strong Engel series and Lüroth series with signs
论文作者
论文摘要
恩格尔系列是$ \ sum_ {j \ geq 1} 1/x_j $的非淘汰序列的正整数的序列$ x_n $，$ x_n $ divides $ x_n $ divides $ x_ {n+1} $ for All $ n \ geq 1 $。在以前的工作中，我们已经证明，对于任何具有更强属性的Engel系列，$ x_n^2 $ divides $ x_ {n+1} $，以$ z_1 = x_1 $和比率$ z_n = x_n = x_n/x_n/x_ {n-1}^2 $ for $ n \ geq for $ n \ geq 2 $。在这里，我们表明，当这种较强的属性拥有时，对于$ \ sum_ {j \ geq 1}ε_j/x_j $，带有任意标志$ε_j= \ pm 1 $的sum $ \ sum_ {j \ geq 1}ε_j/x_j $也是如此。作为应用程序，我们使用此结果为LürothSeries的特定家族和lüroth系列的特定家庭提供明确的持续分数，这些系列由二阶非线性复发定义。我们还为某些由该系列定义的先验数字的家族计算了确切的非理性指数。
An Engel series is a sum of reciprocals $\sum_{j\geq 1} 1/x_j$ of a non-decreasing sequence of positive integers $x_n$ with the property that $x_n$ divides $x_{n+1}$ for all $n\geq 1$. In previous work, we have shown that for any Engel series with the stronger property that $x_n^2$ divides $x_{n+1}$, the continued fraction expansion of the sum is determined explicitly in terms of $z_1=x_1$ and the ratios $z_n=x_n/x_{n-1}^2$ for $n\geq 2$. Here we show that, when this stronger property holds, the same is true for a sum $\sum_{j\geq 1}ε_j/x_j$ with an arbitrary sequence of signs $ε_j=\pm 1$. As an application, we use this result to provide explicit continued fractions for particular families of Lüroth series and alternating Lüroth series defined by nonlinear recurrences of second order. We also calculate exact irrationality exponents for certain families of transcendental numbers defined by such series.
