# The Convergence and Divergence of $q$-Continued Fractions outside the Unit Circle

## TL;DR

This paper investigates the convergence properties of two classes of $q$-continued fractions outside the unit circle, providing theorems that establish conditions for their convergence and that of their odd and even parts.

## Contribution

It introduces new convergence theorems for specific $q$-continued fractions outside the unit circle, expanding understanding of their limit behavior.

## Key findings

- Convergence of $q$-continued fractions outside the unit circle is established.
- Conditions for the convergence of odd and even parts are provided.
- Theorems guarantee convergence at points outside the unit circle.

## Abstract

We consider two classes of $q$-continued fraction whose odd and even parts are limit 1-periodic for $|q|>1$, and give theorems which guarantee the convergence of the continued fraction, or of its odd- and even parts, at points outside the unit circle.