Séminaire Lotharingien de Combinatoire, B18f (1987), 24
pp.
[Formerly: Publ. I.R.M.A. Strasbourg, 1988, 358/S-18, p.
53-76.]
Peter Paule
The Concept of Bailey Chains
Abstract.
In his 1986 book on q-series G. E. Andrews devotes a whole
chapter to Bailey's Lemma and discusses some of its numerous
applications in terms of the Bailey chain concept. The
essence of this concept is an iteration mechanism which
allows to derive a large class of q-series identities by
`reducing' them to more elementary ones. As an example,
the famous Rogers-Ramanujan identities can be reduced to the
q-binomial theorem.
It was G. E. Andrews who observed this iteration mechanism in
its full generality by an appropriate reformulation of
Bailey's Lemma, whereas the author of this survey article
discovered important special cases. W. N. Bailey never formulated
his lemma in that way and consequently missed the full power of
its potential for iteration. Besides introducing the notions of
`Bailey pairs' and `Bailey chains' G. E. Andrews laid the
foundations of a Bailey chain theory for discovering and proving
q-identities.
The purpose of this survey article is to give an introduction
to that concept. Therefore many theorems are not stated in full
generality, for which we refer to the literature.
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