Rogers-Ramanujan-Bailey Identities

The Rogers-Ramanujan identities were first discovered by L.J. Rogers in 1894. These identities, and identities of similar type have fascinated mathematicians for generations, and have found applications in combinatorics, Lie theory, and statistical physics. During the World War II era, W. N. Bailey undertook a careful study of Rogers' work, and in the process simplified and extended it. He discovered what is now known as "Bailey's lemma," the fundamental engine for producing Rogers-Ramanujan type identities, via the insertion of "Bailey pairs." I, in turn, recently undertook a careful study of Bailey's work, and realized that all of the Bailey pairs considered by Bailey are in fact instances of a more general object, which I have named the "standard multiparameter Bailey pair" (SMPBP). Once the SMPBP was identified, it became clear that there were many elegant double-sum Rogers-Ramanujan type identities which had not been previously discovered, which were in some sense "neighbors" of Bailey's own identities. Furthermore, it turns out that many of the identities (both classical and new) which follow as corollaries of the SMPBP can be interpreted combinatorially in terms of a mild generalization of Basil Gordon's famous combinatorial generalization of the Rogers-Ramanujan identities.