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 Rogers 1917 of the Rogers–Ramanujan identities, and Bailey chains were introduced by .
Definition The q-Pochhammer symbols are defined as:
A pair of sequences (αn,βn) is called a Bailey pair if they are related by
or equivalently
Bailey's lemma Bailey's lemma states that if (αn,βn) 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.
References