How to prove Ramanujan's $q$-continued fractions

TL;DR

This paper presents a systematic method based on Euler's approach and Euclid's algorithm to derive Ramanujan's $q$-continued fractions from power series expansions.

Contribution

It introduces a structured technique for deriving Ramanujan's $q$-continued fractions, enhancing understanding and reproducibility of these mathematical objects.

Findings

Systematic derivation of Ramanujan's $q$-continued fractions.

Application of Euclid's algorithm to power series expansions.

Clarification of the connection between power series and continued fractions.

Abstract

By using Euler's approach of using Euclid's algorithm to expand a power series into a continued fraction, we show how to derive Ramanujan's $q$-continued fractions in a systematic manner.