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17 q-Hypergeometric and Related FunctionsProperties

§17.12 Bailey Pairs

Bailey Transform

17.12.1 n=0αnγn=n=0βnδn,


17.12.2 βn =j=0nαjun-jvn+j,
γn =j=nδjuj-nvj+n.

Bailey Pairs

A sequence of pairs of rational functions of several variables (αn,βn), n=0,1,2,, is called a Bailey pair provided that for each n0

17.12.3 βn=j=0nαj(q;q)n-j(aq;q)n+j.

Weak Bailey Lemma

If (αn,βn) is a Bailey pair, then

17.12.4 n=0qn2anβn=1(aq;q)n=0qn2anαn.

Strong Bailey Lemma

If (αn,βn) is a Bailey pair, then so is (αn,βn), where

17.12.5 (aqρ1,aqρ2;q)nαn =(ρ1,ρ2;q)n(aqρ1ρ2)nαn
(aqρ1,aqρ2;q)nβn =j=0n(ρ1,ρ2;q)j(aqρ1ρ2;q)n-j(aqρ1ρ2)jβj(q;q)n-j

When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain.

The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is:

17.12.6 αn =(a;q)n(1-aq2n)(-1)nqn(3n-1)/2an(q;q)n(1-a),
βn =1(q;q)n.

The Bailey pair and Bailey chain concepts have been extended considerably. See Andrews (2000, 2001), Andrews and Berkovich (1998), Andrews et al. (1999), Milne and Lilly (1992), Spiridonov (2002), and Warnaar (1998).