Two complementary relations for the Rogers-Ramanujan continued fraction

TL;DR

This paper provides new proofs for two complementary relations involving the Rogers-Ramanujan continued fraction, utilizing product expansions of Jacobi theta functions, thus offering alternative insights into these classical identities.

Contribution

It introduces novel proofs for Ramanujan's relations for R(q) using only Jacobi theta function product expansions, simplifying previous approaches.

Findings

Established two complementary relations for R(q)

Provided proofs solely based on Jacobi theta function products

Enhanced understanding of Rogers-Ramanujan continued fraction identities

Abstract

Let $R(q)$ be the Rogers-Ramanujan continued fraction. We give different proofs of two complementary relations for $R(q)$ given by Ramanujan and proved by Watson and Ramanathan. Our proofs only use product expansions for classical Jacobi theta functions.