On identities of the Rogers--Ramanujan type

Abstract

A generalized Bailey pair, which contains several special cases considered by Bailey (\emph{Proc. London Math. Soc. (2)}, 50 (1949), 421--435), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of associated $q$-difference equations points to a connection with a mild extension of Gordon's combinatorial generalization of the Rogers-Ramanujan identities (\emph{Amer. J. Math.}, 83 (1961), 393--399). This, in turn, allows the formulation of natural combinatorial interpretations of many of the identities in Slater's list (\emph{Proc. London Math. Soc. (2)} 54 (1952), 147--167), as well as the new identities presented here. A list of 26 new double sum--product Rogers-Ramanujan type identities are included as an appendix.