§17.12 Bailey Pairs

Bailey Transform

 17.12.1 $\sum_{n=0}^{\infty}\alpha_{n}\gamma_{n}=\sum_{n=0}^{\infty}\beta_{n}\delta_{n},$ ⓘ Symbols: $n$: nonnegative integer, $\alpha_{n}$: part of Bailey pair and $\beta_{n}$: part of Bailey pair Permalink: http://dlmf.nist.gov/17.12.E1 Encodings: TeX, pMML, png See also: Annotations for §17.12, §17.12 and Ch.17

where

 17.12.2 $\displaystyle\beta_{n}$ $\displaystyle=\sum_{j=0}^{n}\alpha_{j}u_{n-j}v_{n+j},$ $\displaystyle\gamma_{n}$ $\displaystyle=\sum_{j=n}^{\infty}\delta_{j}u_{j-n}v_{j+n}.$ ⓘ Symbols: $j$: nonnegative integer, $n$: nonnegative integer, $\alpha_{n}$: part of Bailey pair and $\beta_{n}$: part of Bailey pair Permalink: http://dlmf.nist.gov/17.12.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §17.12, §17.12 and Ch.17

Bailey Pairs

A sequence of pairs of rational functions of several variables $(\alpha_{n},\beta_{n})$, $n=0,1,2,\dots$, is called a Bailey pair provided that for each $n\geqq 0$

 17.12.3 $\beta_{n}=\sum_{j=0}^{n}\frac{\alpha_{j}}{\left(q;q\right)_{n-j}\left(aq;q% \right)_{n+j}}.$

Weak Bailey Lemma

If $(\alpha_{n},\beta_{n})$ is a Bailey pair, then

 17.12.4 $\sum_{n=0}^{\infty}q^{n^{2}}a^{n}\beta_{n}=\frac{1}{\left(aq;q\right)_{\infty}% }\sum_{n=0}^{\infty}q^{n^{2}}a^{n}\alpha_{n}.$

Strong Bailey Lemma

If $(\alpha_{n},\beta_{n})$ is a Bailey pair, then so is $(\alpha_{n}^{\prime},\beta_{n}^{\prime})$, where

 17.12.5 $\displaystyle\left(\frac{aq}{\rho_{1}},\frac{aq}{\rho_{2}};q\right)_{n}\alpha_% {n}^{\prime}$ $\displaystyle=\left(\rho_{1},\rho_{2};q\right)_{n}\left(\frac{aq}{\rho_{1}\rho% _{2}}\right)^{n}\alpha_{n}$ $\displaystyle\left(\frac{aq}{\rho_{1}},\frac{aq}{\rho_{2}};q\right)_{n}\beta_{% n}^{\prime}$ $\displaystyle=\sum_{j=0}^{n}\left(\rho_{1},\rho_{2};q\right)_{j}\left(\frac{aq% }{\rho_{1}\rho_{2}};q\right)_{n-j}\left(\frac{aq}{\rho_{1}\rho_{2}}\right)^{j}% \frac{\beta_{j}}{\left(q;q\right)_{n-j}}$

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 $\displaystyle\alpha_{n}$ $\displaystyle=\frac{\left(a;q\right)_{n}(1-aq^{2n})(-1)^{n}q^{n(3n-1)/2}a^{n}}% {\left(q;q\right)_{n}(1-a)},$ $\displaystyle\beta_{n}$ $\displaystyle=\frac{1}{\left(q;q\right)_{n}}.$

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).