# On the Divergence in the General Sense of $q$-Continued Fraction on the Unit Circle

## TL;DR

This paper demonstrates that within a specific class of $q$-continued fractions, including famous examples like Rogers-Ramanujan, there are uncountably many points on the unit circle where these fractions diverge in the general sense, impacting their convergence understanding.

## Contribution

It establishes the existence of uncountably many divergence points on the unit circle for a class of $q$-continued fractions, including notable examples like Rogers-Ramanujan.

## Key findings

- Uncountably many divergence points on the unit circle.
- Includes key continued fractions like Rogers-Ramanujan.
- Implications for convergence of other $q$-continued fractions.

## Abstract

We show, for each $q$-continued fraction $G(q)$ in a certain class of continued fractions, that there is an uncountable set of points on the unit circle at which $G(q)$ diverges in the general sense. This class includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fraction. We discuss the implications of our theorems for the general convergence of other $q$-continued fractions, for example the G\"{o}llnitz-Gordon continued fraction, on the unit circle.