This is the author’s first formal treatment of "Bailey chains", a technique for discovering and proving q-series identities which has proven to be very powerful. A "Bailey pair" is a pair of sequences satisfying
W. N. Bailey [Proc. Lond. Math. Soc., II. Ser. 49, 421- 435 (1947; Zbl 0041.034)] showed that any q-series identity which defines a Bailey pair can be transformed into a new identity. It is the author’s observation that each Bailey pair in fact lies inside a doubly infinite sequence of Bailey pairs, any one of which implies the others. Thus any identity of the form of (*) gives rise to a doubly infinite family of identities. For example, the 130 identities of L. J. Slater [ibid. 54, 147-167 (1952; Zbl 0046.272)] become 130 distinct infinite families of identities. More usefully, given a conjectured identity of type (*), it is possible to move up or down in the Bailey chain in hopes of finding an equivalent Bailey pair for which the identity is easier to prove. This is essentially what L. J. Rogers did to prove the Rogers-Ramanujan identities.