Convergence properties of the classical and generalized Rogers-Ramanujan continued fraction

TL;DR

This paper investigates the convergence and divergence behavior of Rogers-Ramanujan continued fractions on the unit circle, providing new examples and conditions for convergence at roots of unity, extending classical results.

Contribution

It introduces an uncountable measure-zero set where the Rogers-Ramanujan continued fraction diverges and establishes explicit convergence/divergence criteria for generalized fractions at roots of unity.

Findings

Identified an uncountable measure-zero set of divergence.

Derived explicit convergence conditions for generalized fractions at roots of unity.

Extended classical results of Schur and Ramanujan on convergence behaviors.

Abstract

The aim of this paper is to study the convergence and divergence of the Rogers-Ramanujan and the generalized Rogers-Ramanujan continued fractions on the unit circle. We provide an example of an uncountable set of measure zero on which the Rogers-Ramanujan continued fraction $R(x)$ diverges and which enlarges a set previously found by Bowman and Mc Laughlin. We further study the generalized Rogers-Ramanujan continued fractions $R_a(x)$ for roots of unity $a$ and give explicit convergence and divergence conditions. As such, we extend some work of Huang towards a question originally investigated by Ramanujan and some work of Schur on the convergence of $R(x)$ at roots of unity. In the end, we state several conjectures and possible directions for generalizing Schur's result to all Rogers-Ramanujan continued fractions $R_a(x)$.