Two modular equations for squares of the Rogers–Ramanujan functions with applications

Abstract

Two modular identities of Gordon, McIntosh, and Robins are shown to be connected to the Rogers–Ramanujan continued fraction R(q), and in particular to Ramanujan’s parameter k:=R(q)R2(q2). Using this connection, we give new modular relations for R(q), and offer new and uniform proofs of several results of Ramanujan. In particular, we give a new proof of a famous and fundamental modular identity satisfied by the Rogers–Ramanujan continued fraction. We furthermore show that many analogous results hold for Ramanujan’s parameters μ:=R(q)R(q4) and ν:=R2(q1/2)R(q)/R(q2). New proofs are offered for modular relations connecting R(q) to R(−q), R(q2), and R(q4), and new relations connecting R(q) at these arguments are offered. Eleven identities for the Rogers–Ramanujan functions are proved, including four new identities.