# The Convergence Behavior of $q$-Continued Fractions on the Unit Circle

## TL;DR

This paper extends previous results on the divergence of the Rogers-Ramanujan continued fraction to a broader class of $q$-continued fractions, showing uncountably many points on the unit circle where these fractions do not converge.

## Contribution

It generalizes the divergence results to a wider class of $q$-continued fractions, including several famous examples, and discusses implications for other related fractions.

## Key findings

- Uncountable sets of points on the unit circle where $q$-continued fractions diverge.
- Generalization of divergence results from Rogers-Ramanujan to broader classes.
- Implications for convergence behavior of other $q$-continued fractions.

## Abstract

In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of $q$-continued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each $q$-continued fraction, $G(q)$, in this class, that there is an uncountable set of points, $Y_{G}$, on the unit circle such that if $y \in Y_{G}$ then $G(y)$ does not converge to a finite value. We discuss the implications of our theorems for the convergence of other $q$-continued fractions, for example the G\"ollnitz-Gordon continued fraction, on the unit circle.