In mathematics, a Bailey pair is a pair of sequences satisfying certain relations, and a Bailey chain is a sequence of Bailey pairs. Bailey pairs were introduced by while studying
the second proof of the Rogers-Ramanujan identities

Rogers-Ramanujan identities

In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series, first discovered and proved by . They were subsequently rediscovered by Srinivasa Ramanujan some time before 1913. Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they...

, and Bailey chains were introduced by .

Definition

The q-Pochhammer symbols are defined as:


A pair of sequences (α_n,β_n) is called a Bailey pair if they are related by
or equivalently

Bailey's lemma

Bailey's lemma states that if (α_n,β_n) is a Bailey pair, then so is
(α'_n,β'_n) where
In other words, given one Bailey pair, one can construct a second using the formulas above. This process can be iterated to produce an infinite sequence of Bailey pairs, called a Bailey chain.

Examples

An example of a Bailey pair is given by

gave a list of 130 examples related to Bailey pairs.