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 In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by , and were subsequently rediscovered (without a proof) by Srin ...

, and Bailey chains were introduced by .

Definition

The

q-Pochhammer symbol In the mathematical field of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer ...

(a;q)_n

are defined as: :

(a;q)_n = \prod_(1-aq^j) = (1-a)(1-aq)\cdots(1-aq^).

A pair of sequences (α_''n'',β_''n'') is called a Bailey pair if they are related by :

\beta_n=\sum_^n\frac

or equivalently :

\alpha_n = (1-aq^)\sum_^n\frac.

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 :

\alpha^\prime_n= \frac

\beta^\prime_n = \sum_\frac.

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 :

\alpha_n = q^\sum_^n(-1)^jq^, \quad \beta_n = \frac.

gave a list of 130 examples related to Bailey pairs.

References

* * * * * * *{{Citation , last1=Warnaar , first1=S. Ole , title=Algebraic combinatorics and applications (Gössweinstein, 1999) , url=http://www.maths.uq.edu.au/~uqowarna/pubs/Bailey50.pdf , publisher=Springer-Verlag , location=Berlin, New York , mr=1851961 , year=2001 , chapter=50 years of Bailey's lemma , pages=333–347 Special functions Q-analogs