Continued fractions for some transcendental numbers

TL;DR

This paper investigates the continued fraction expansions of certain transcendental numbers formed by series with terms satisfying a specific recurrence relation, revealing interlaced patterns involving sequence terms and ratios.

Contribution

It introduces a novel connection between non-autonomous recurrence sequences and the continued fraction expansions of associated transcendental sums.

Findings

Sequence terms and ratios appear interlaced in the continued fraction expansion.

The sum of the series is proven to be transcendental.

Patterns in continued fractions relate to the recurrence structure.

Abstract

We consider series of the form $$ \frac{p}{q} +\sum_{j=2}^\infty \frac{1}{x_j}, $$ where $x_1=q$ and the integer sequence $(x_n)$ satisfies a certain non-autonomous recurrence of second order, which entails that $x_n|x_{n+1}$ for $n\geq 1$. It is shown that the terms of the sequence, and multiples of the ratios of successive terms, appear interlaced in the continued fraction expansion of the sum of the series, which is a transcendental number.