Elementary derivations of the Rogers-Fine identity and other q-series identities

We begin the article with a proof of the Rogers-Fine identity. We then show that the Rogers-Fine identity implies the Rogers-Ramanujan identities as well as a new finite version of the quintuple identity. Motivated by the connections between these identities, we discover an identity which yields proofs of Rogers-Ramanujan-type identities associated with the Rogers-Ramanujan continued fraction, the Ramanujan-Göllnitz-Gordon continued fraction and Ramanujan's cubic continued fraction. We also discover a new generalization of the quintuple product identity which leads to a generalization of an identity due to R.J. Evans and a short proof of q-Chu-Vandermonde identity that does not require the knowledge of the q-binomial theorem.