The Rogers--Ramanujan continued fraction
Bruce C. Berndt, Örs Rebák
TL;DR
The paper surveys the Rogers–Ramanujan continued fraction $R(q)$, its connections to the Rogers–Ramanujan identities, and its rich structure inside $q$-series, theta functions, and modular forms. It synthesizes definitions, convergence, and boundary behavior, along with Ramanujan’s evaluations at special $q$, class-invariant techniques, and quintic analogues for $R(q)$ and $R(q^4)$, tying these to the identities $G(q)$ and $H(q)$. It also catalogs a web of exact identities involving $R(q)$, including factorization formulas, two-variable theta-function relations, and Ramanujan’s $k$-parameter framework, as well as finite forms and an unusual identity from Ramanujan’s notebooks. The historical interludes recount Ramanujan’s notebooks and the Lost Notebook, their discovery, and their lasting influence on the field. Overall, the work provides a cohesive panorama of analytic, algebraic, and combinatorial facets surrounding $R(q)$ and its place in Ramanujan’s legacy, with emphasis on explicit evaluations and structural identities that connect to partitions, class invariants, and modular phenomena.
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.