Continued fractions for strong Engel series and L\"{u}roth series with signs

Abstract

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 $\sum_{j\geq 1}\epsilon_j/x_j$ with an arbitrary sequence of signs $\epsilon_j=\pm 1$. As an application, we use this result to provide explicit continued fractions for particular families of L\"{u}roth series and alternating L\"{u}roth series defined by nonlinear recurrences of second order. We also calculate exact irrationality exponents for certain families of transcendental numbers defined by such series.