50 Years of Bailey’s Lemma

Half a century ago, The Proceedings of the London Mathematical Society published W. N. Bailey’s influential paper Identities of the Rogers–Ramanujan type [16]. The main result therein, which was inspired by Rogers’ second proof of the Rogers– Ramanujan identities [49] (and also [48, 28, 15]), is what is now known as Bailey’s lemma. To celebrate the occasion of the lemma’s fiftieth birthday we present a history of Bailey’s lemma in 5 chapters (or rather sections), covering (i) Bailey’s work, (ii) the Bailey chain (iii) the Bailey lattice (iv) the Bailey lemma in statistical mechanics, and (v) conjugate Bailey pairs. Due to size limitations of this paper the higher rank [42, 40, 43, 41, 14, 60] and trinomial [11, 59, 19] generalizations of the Bailey lemma will be treated at the lemma’s centennial in 2049. More extensive reviews of topics (i), (ii) and (iii), can be found in [5, Sec. 3], [46] and [24], respectively.