Continued fractions for strong Engel series and Lüroth series with signs
Andrew N. W. Hone, Juan Luis Varona
TL;DR
The paper extends explicit continued fraction representations from strong Engel series to sums with arbitrary signs, under the strong Engel condition $x_n^2 \mid x_{n+1}$, by expressing the CF of $S=\frac{p}{q}+\sum_{j\ge2}\frac{\epsilon_j}{x_j}$ as iterative foldings using $\varphi_{z_j}^{(\epsilon_j)}$ on the base CF given by $[a_0;\mathbf{a}]$ with $z_j=x_j/x_{j-1}^2$. It then applies this folding framework to Lüroth-type and alternating Lüroth series defined by second-order recurrences, deriving explicit foldings and CFs for broad sign-variant families, and providing exact irrationality-exponent results for selected cases. The work establishes transcendence results (often via $\mu>2$) for these series and conjectures transcendence for all strong Engel series with signs, tying together continued fractions, Engel/Pierce representations, and nonlinear recurrences. These contributions yield explicit Diophantine properties and convergent approximants for a wide class of sign-weighted Engel and Lüroth-type series.
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 $\sum_{j\geq 1}ε_j/x_j$ 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.