# A $q$-continued fraction

## TL;DR

This paper employs generating functions to evaluate a four-parameter q-continued fraction, providing new proofs of classical identities like Ramanujan's and deriving generalizations of known continued fractions.

## Contribution

It introduces a novel method to find the limit of a q-continued fraction as a ratio of q-series and applies this to reprove and extend important identities.

## Key findings

- Derived the limit of a four-parameter q-continued fraction.
- Provided new proofs of Ramanujan's continued fraction identities.
- Generalized Ramanujan's continued fractions.

## Abstract

We use the method of generating functions to find the limit of a $q$-continued fraction, with 4 parameters, as a ratio of certain $q$-series. We then use this result to give new proofs of several known continued fraction identities, including Ramanujan's continued fraction expansions for $(q^2;q^3)_{\infty}/(q;q^3)_{\infty}$ and $(q;q^2)_\infty / (q^{3};q^{6})_\infty^3$. In addition, we give a new proof of the famous Rogers-Ramanujan identities. We also use our main result to derive two generalizations of another continued fraction due to Ramanujan.