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 .
The q-Pochhammer symbols
(a;q)n
(a;q)n=\prod0\le(1-aqj)=(1-a)(1-aq) … (1-aqn-1).
A pair of sequences (αn,βn) is called a Bailey pair if they are related by
\betan=\sum
| ||||
r=0 |
\alphan=(1-aq2n
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)\sum | ||||
j=0 |
.
Bailey's lemma states that if (αn,βn) is a Bailey pair, then so is (α'n,β'n) where
\prime | |
\alpha | |
n= |
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(aq/\rho1;q)n(aq/\rho2;q)n |
\prime | |
\beta | |
n |
=\sumj\ge0
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(q;q)n-j(aq/\rho1;q)n(aq/\rho2;q)n |
.
An example of a Bailey pair is given by
\alphan=
n2+n | |
q |
n(-1) | |
\sum | |
j=-n |
jq
-j2 | |
, \betan=
(-q)n | ||||||||||||
|
.
gave a list of 130 examples related to Bailey pairs.