Continued fractions for strong Engel series and Lüroth series with signs

Abstract

An Engel series is a sum of reciprocals ${\sum }_{j\ge 1}1/x_j$ of a non-decreasing sequence of positive integers $x_n$ with the property that $x_n$ divides $x_{n+1}$ for all $n\ge 1$. 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 $n\ge 2$. Here we show that, when this stronger property holds, the same is true for a sum ${\sum }_{j\ge 1}{\epsilon }_j/x_j$ 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 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.