On the Equivalence of Ramanujan's Partition Identities and a Connection with the Rogers–Ramanujan Continued Fraction

Abstract

A famous identity of Ramanujan connected with partitions modulo 5 is shown to be equivalent to another identity of Ramanujan. The latter identity is used to establish a differential equation for the Rogers–Ramanujan continued fraction found in Ramanujan's lost notebook. We also prove that two other identities of Ramanujan are equivalent, one of which is associated with Ramanujan's partition congruence modulo 7. Last, we give a new proof of the transformation formula for the Dedekind eta-function, which is used in our proofs of equivalence.