Ramanujan's $q$-continued fractions

TL;DR

This paper reviews and reorganizes Ramanujan's $q$-continued fractions, highlighting their significance in basic hypergeometric series and summarizing contributions from various authors.

Contribution

It provides a comprehensive summary and re-organization of Ramanujan's $q$-continued fractions, clarifying their structure and historical development.

Findings

Reorganization of Ramanujan's $q$-continued fractions

Summary of contributions from multiple authors

Clarification of their role in hypergeometric series

Abstract

Ramanujan's $q$-continued fractions are a central part of Ramanujan's development of basic hypergeometric series. They appear in Chapter 16 of Part III and Chapter 32 of Part V of {\em Ramanujan's Notebooks} edited by Berndt, and in Volume I of Andrews and Berndt's {\em Ramanujan's Lost Notebook}. In these references the continued fractions as presented in the order in which they appear in Ramanujan's original notebooks. We summarize the work of several authors on this topic and re-organize Ramanujan's $q$-continued fractions.