Elementary proofs of some identities of Ramanujan for the Rogers–Ramanujan functions

In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers–Ramanujan functions. With one exception all of Ramanujanʼs identities were proved. In this paper, we provide a proof for the remaining identity together with new elementary proofs for two identities of Ramanujan which were previously proved using the theory of modular forms. Ramanujan stated that each of his formula was the simplest of a large class. Our proofs are constructive and permit us to obtain several analogous identities which could have been stated by Ramanujan and may very well belong to his large class of identities.