History and Development of Bailey's Lemma

Abstract

The seminal discovery in the theory of q-series and combinatorics analysis has been the
Bailey lemma which has since made a revolutionary impact on the study of partition identities,
modular forms and special functions since it was presented by W.N. Bailey in the middle of
the 20th century. This review article has tried to give the historical details of the Bailey lemma
and recounts the origin, evolution and subsequent generalizations of the lemma in noting the
massive contribution to a wide range of mathematics including number theory, representation
theory, and the physics of mathematics. This paper starts with an introduction to the classical
version of the Bailey lemma, including how it was used in the first few years of its discovery to
provide systematic proofs of Rogers-Ramanujan-type identities and other forms of partitions. It
later explores the methodological development of the lemma which was laid down by important
authors like George Andrews and Basil Gordon, who introduced the notion of Bailey chains
as well as Bailey pairs as expansions of the work of Bailey. The developments have allowed us
to discover new infinite families of identities and also achieve a better understanding of the
underlying combinatorial structures. Recent progress, including elliptic and multidimensional
extensions of the Bailey lemma and their application to vertex operator algebras and conformal
field theory are also examined in the paper. In a bid to rekindle the perpetual relevance and
applicability of the Bailey lemma as a unifying tool in mathematical studies, this paper aims
at uniting historical and modern perspectives of the same. The envisaged outcomes are the
improved insight into the flexibility of the lemma, the interconnections with other branches
of mathematics, and many more opportunities to be pursued in the further studies, especially
following the explosion of interdisciplinary applications of the lemma.