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17 q-Hypergeometric and Related FunctionsProperties

Β§17.12 Bailey Pairs

Bailey Transform

17.12.1 βˆ‘n=0∞αn⁒γn=βˆ‘n=0∞βn⁒δn,


17.12.2 Ξ²n =βˆ‘j=0nΞ±j⁒unβˆ’j⁒vn+j,
Ξ³n =βˆ‘j=n∞δj⁒ujβˆ’n⁒vj+n.

Bailey Pairs

A sequence of pairs of rational functions of several variables (Ξ±n,Ξ²n), n=0,1,2,…, is called a Bailey pair provided that for each n≧0

17.12.3 Ξ²n=βˆ‘j=0nΞ±j(q;q)nβˆ’j⁒(a⁒q;q)n+j.

Weak Bailey Lemma

If (Ξ±n,Ξ²n) is a Bailey pair, then

17.12.4 βˆ‘n=0∞qn2⁒an⁒βn=1(a⁒q;q)βˆžβ’βˆ‘n=0∞qn2⁒an⁒αn.

Strong Bailey Lemma

If (Ξ±n,Ξ²n) is a Bailey pair, then so is (Ξ±nβ€²,Ξ²nβ€²), where

17.12.5 (a⁒qρ1,a⁒qρ2;q)n⁒αnβ€² =(ρ1,ρ2;q)n⁒(a⁒qρ1⁒ρ2)n⁒αn
(a⁒qρ1,a⁒qρ2;q)n⁒βnβ€² =βˆ‘j=0n(ρ1,ρ2;q)j⁒(a⁒qρ1⁒ρ2;q)nβˆ’j⁒(a⁒qρ1⁒ρ2)j⁒βj(q;q)nβˆ’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 Β§17.2(vi) is:

17.12.6 Ξ±n =(a;q)n⁒(1βˆ’a⁒q2⁒n)⁒(βˆ’1)n⁒qn⁒(3⁒nβˆ’1)/2⁒an(q;q)n⁒(1βˆ’a),
Ξ²n =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).