Continued fractions for strong Engel series and Lüroth series with signs

An Engel series is a sum of reciprocals ∑j≥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≥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≥2. Here we show that, when this stronger property holds, the same is true for a sum ∑j≥1 ϵ_j/x_j with an arbitrary sequence of signs ϵ_j=±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.