The Rogers--Ramanujan continued fraction

TL;DR

This paper surveys the properties, identities, and generalizations of the Rogers--Ramanujan continued fraction, highlighting its historical context and connections to Ramanujan's work and identities.

Contribution

It compiles and reviews known properties, identities, and generalizations of the Rogers--Ramanujan continued fraction, emphasizing historical insights from Ramanujan's letters and notebooks.

Findings

Comprehensive overview of Rogers--Ramanujan continued fraction properties

Connections to Ramanujan's identities and historical documents

Summary of generalizations and related identities

Abstract

The primary purpose of this paper is to provide a survey of properties, values, identities, and generalizations of the Rogers--Ramanujan continued fraction, which is closely related to the Rogers--Ramanujan identities. Many of these results are found in Ramanujan's first two letters to Hardy, Ramanujan's notebooks, and his lost notebook. Short historical accounts are provided for both the notebooks and lost notebook.