Bailey pair

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 W. N. Bailey (1947, 1948) while studying the second proof Rogers 1917 of the Rogers–Ramanujan identities, and Bailey chains were introduced by .

Definition

The q-Pochhammer symbols $(a;q{)}_n$ are defined as:

$(a;q{)}_n=\prod _{0\le j<n}(1-a{q}^j)=(1-a)(1-aq)\cdots (1-a{q}^{n-1}).$

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

${\beta }_n=\sum _{r=0}^n\frac{{\alpha }_r}{(q;q{)}_{n-r}(aq;q{)}_{n+r}}$

or equivalently

${\alpha }_n=(1-a{q}^{2n})\sum _{j=0}^n\frac{(aq;q{)}_{n+j-1}(-1{)}^{n-j}{q}^{(\genfrac{}{}{0ex}{}{n-j}{2})}{\beta }_j}{(q;q{)}_{n-j}}.$

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

${\alpha }_n^{\prime }=\frac{({\rho }_1;q{)}_n({\rho }_2;q{)}_n(aq/{\rho }_1{\rho }_2{)}^n{\alpha }_n}{(aq/{\rho }_1;q{)}_n(aq/{\rho }_2;q{)}_n}$

${\beta }_n^{\prime }=\sum _{j\ge 0}\frac{({\rho }_1;q{)}_j({\rho }_2;q{)}_j(aq/{\rho }_1{\rho }_2;q{)}_{n-j}(aq/{\rho }_1{\rho }_2{)}^j{\beta }_j}{(q;q{)}_{n-j}(aq/{\rho }_1;q{)}_n(aq/{\rho }_2;q{)}_n}.$

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

${\alpha }_n=q^{n^2+n}\sum _{j=-n}^n(-1{)}^j{q}^{-j^2},{\beta }_n=\frac{(-q{)}^n}{(q^2;q^2{)}_n}.$

L. J. Slater (1952) gave a list of 130 examples related to Bailey pairs.