# Bailey Transform

 17.12.1 $\sum_{n=0}^{\infty}\alpha_{n}\gamma_{n}=\sum_{n=0}^{\infty}\beta_{n}\delta_{n},$

where

 17.12.2 $\displaystyle\beta_{n}$ $\displaystyle=\sum_{j=0}^{n}\alpha_{j}u_{n-j}v_{n+j},$ $\displaystyle\gamma_{n}$ $\displaystyle=\sum_{j=n}^{\infty}\delta_{j}u_{j-n}v_{j+n}.$

# 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).