Generalizations of some $q$-product Identities of Ramanujan and others

Abstract: By considering certain limiting cases of a WP-Bailey chain discovered by Andrews, and also limiting cases of certain classical summation formulae for basic hypergeometric series, we derive new expressions for certain Lambert series in terms of basic hypergeometric series. In some cases, the resulting series involve an arbitrary Bailey pair. This allows for the derivation of new basic hypergeometric expansions for some $q$-products and series that Ramanujan expressed in terms of Lambert series. Some of Ramanujan’s identities are extended to more general relations containing one or more free parameters.