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The concept of Bailey chains. (English) Zbl 0978.05521
Summary: 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.

MSC:
 05A30 $$q$$-calculus and related topics 05A19 Combinatorial identities, bijective combinatorics 05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
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