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 Template:Harvs while studying the second proof Rogers 1917 of the Rogers–Ramanujan identities, and Bailey chains were introduced by Template:Harvtxt.

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 Template:Harv

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

Template:Harvs gave a list of 130 examples related to Bailey pairs.

References

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