§17.12 Bailey Pairs

Notes:: See Andrews (1984).
Referenced by:: §17.17
Permalink:: http://dlmf.nist.gov/17.12

¶ Bailey Transform

Keywords:: $q$ -hypergeometric function

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

where

17.12.2

$\beta _{n}=\sum _{{j=0}}^{n}\alpha _{j}u_{{n-j}}v_{{n+j}},$

$\gamma _{n}=\sum _{{j=n}}^{{\infty}}\delta _{j}u_{{j-n}}v_{{j+n}}.$

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

¶ Bailey Pairs

Keywords:: $q$ -hypergeometric function

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 $\beta _{n}=\sum _{{j=0}}^{n}\frac{\alpha _{j}}{\left(q;q\right)_{{n-j}}\left(aq;q\right)_{{n+j}}}.$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $j$ : 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.E3
Encodings:: TeX, pMML, png

¶ Weak Bailey Lemma

Keywords:: $q$ -hypergeometric function

If $(\alpha _{n},\beta _{n})$ is a Bailey pair, then

17.12.4 $\sum _{{n=0}}^{{\infty}}q^{{n^{2}}}a^{n}\beta _{n}=\frac{1}{\left(aq;q\right)_{{\infty}}}\sum _{{n=0}}^{{\infty}}q^{{n^{2}}}a^{n}\alpha _{n}.$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $\alpha _{n}$ : part of Bailey pair and $\beta _{n}$ : part of Bailey pair
Permalink:: http://dlmf.nist.gov/17.12.E4
Encodings:: TeX, pMML, png

¶ Strong Bailey Lemma

Keywords:: Rogers–Ramanujan identities, $q$ -hypergeometric function

If $(\alpha _{n},\beta _{n})$ is a Bailey pair, then so is $(\alpha _{n}^{{\prime}},\beta _{n}^{{\prime}})$ , where

17.12.5

$\left(\frac{aq}{\rho _{1}},\frac{aq}{\rho _{2}};q\right)_{{n}}\alpha _{n}^{{\prime}}=\left(\rho _{1},\rho _{2};q\right)_{{n}}\left(\frac{aq}{\rho _{1}\rho _{2}}\right)^{n}\alpha _{n}$

$\left(\frac{aq}{\rho _{1}},\frac{aq}{\rho _{2}};q\right)_{{n}}\beta _{n}^{{\prime}}=\sum _{{j=0}}^{n}\left(\rho _{1},\rho _{2};q\right)_{{j}}\left(\frac{aq}{\rho _{1}\rho _{2}};q\right)_{{n-j}}\left(\frac{aq}{\rho _{1}\rho _{2}}\right)^{j}\frac{\beta _{j}}{\left(q;q\right)_{{n-j}}}$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $j$ : nonnegative integer, $n$ : nonnegative integer, $\alpha _{n}$ : part of Bailey pair and $\beta _{n}$ : part of Bailey pair
Referenced by:: ¶ ‣ §17.12
Permalink:: http://dlmf.nist.gov/17.12.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

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

$\alpha _{n}=\frac{\left(a;q\right)_{{n}}(1-aq^{{2n}})(-1)^{n}q^{{n(3n-1)/2}}a^{n}}{\left(q;q\right)_{{n}}(1-a)},$

$\beta _{n}=\frac{1}{\left(q;q\right)_{{n}}}.$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $\alpha _{n}$ : part of Bailey pair and $\beta _{n}$ : part of Bailey pair
Permalink:: http://dlmf.nist.gov/17.12.E6
Encodings:: TeX, TeX, pMML, pMML, png, png

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).