Generalized continued fraction expansions for $\pi$ and $e$

TL;DR

This paper proves some conjectures on continued fraction expansions of fundamental constants like e, introduced by recent machine learning-based conjectures, and presents a method to generate new continued fractions from series representations.

Contribution

It verifies conjectures on continued fractions of e and introduces a method to generate continued fractions from series representations.

Findings

Proved several conjectures on continued fractions of e.

Established equivalence among some of the conjectures.

Proposed a simple method to generate continued fractions from series.

Abstract

Recently Raayoni et al. announced various conjectures on continued fractions of fundamental constants automatically generated with machine learning techniques. In this paper we prove some of their stated conjectures for Euler number $e$ and show the equivalence of some of the listed conjectures. Moreover, we propose a simple method that can be used to generate other continued fractions using their series representations.