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

TL;DR

This paper extends previous work on continued fractions for Engel series by including alternating signs, providing explicit formulas for L"uroth series and their irrationality exponents, thus advancing the understanding of these series' properties.

Contribution

It generalizes the continued fraction expansion results to series with arbitrary signs and applies this to L"uroth series, including nonlinear recurrences, with explicit formulas and irrationality exponents.

Findings

Explicit continued fractions for signed Engel series

Application to L"uroth series with nonlinear recurrences

Calculation of irrationality exponents for transcendental 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 $\sum_{j\geq 1}\epsilon_j/x_j$ 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.