Some theorems on the Rogers-Ramanujan continued fraction and associated theta function identities in Ramanujan's lost notebook

In his lost notebook, Ramanujan recorded several modular equations of degree 5 related to the Rogers-Ramanujan continued fraction R(q). We prove several of these identities and give factorizations of some of them in this paper. The parameter k = R(q) R-2(q(2)) introduced by Ramanujan in his second notebook has not been recognized for its usefulness. In this work, we demonstrate how beautifully the parameter k works, as we prove several identities involving k stated by Ramanujan in the lost notebook.