On the continued fraction expansion of certain Engel series

TL;DR

This paper derives explicit continued fraction expansions for a specific class of Engel series where each term is divisible by the square of the previous, including examples linked to nonlinear recurrences and transcendental numbers.

Contribution

It provides a new explicit formula for continued fractions of a special Engel series class, extending previous results and exploring related transcendental numbers.

Findings

Explicit continued fraction expressions derived for the series

Includes series with known expansions by Shallit

Analyzes examples from nonlinear recurrences with the Laurent property

Abstract

An Engel series is a sum of the reciprocals of an increasing sequence of positive integers, which is such that each term is divisible by the previous one. Here we consider a particular class of Engel series, for which each term of the sequence is divisible by the square of the preceding one, and find an explicit expression for the continued fraction expansion of the sum of a generic series of this kind. As a special case, this includes certain series whose continued fraction expansion was found by Shallit. A family of examples generated by nonlinear recurrences with the Laurent property is considered in detail, along with some associated transcendental numbers.