A generalization of a modular identity of Rogers

In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers–Ramanujan functions. Most of the elementary proofs given for these identities are based on Schröter-type theta function identities in particular, the identities of L.J. Rogers. We give a generalization of Rogers's identity that also generalizes similar formulas of H. Schröter, and of R. Blecksmith, J. Brillhart, and I. Gerst. Applications to modular equations, Ramanujan's identities for the Rogers–Ramanujan functions as well as new identities for these functions are given.