17 q-Hypergeometric and Related Functions Properties 17.11 Transformations of

q

§17.12 Bailey Pairs

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Notes:: See Andrews (1984).
Referenced by:: §17.17
Permalink:: http://dlmf.nist.gov/17.12
See also:: Annotations for Ch.17

Bailey Transform

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Keywords:: Bailey transform, $q$ -hypergeometric function
See also:: Annotations for §17.12 and Ch.17

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

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Keywords:: Bailey pairs, $q$ -hypergeometric function
See also:: Annotations for §17.12 and Ch.17

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: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (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 See also: Annotations for §17.12, §17.12 and Ch.17

Weak Bailey Lemma

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Keywords:: Bailey lemma, $q$ -hypergeometric function, weak
See also:: Annotations for §17.12 and Ch.17

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: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (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 See also: Annotations for §17.12, §17.12 and Ch.17

Strong Bailey Lemma

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Keywords:: Bailey chain, Bailey lemma, Rogers–Ramanujan identities, $q$ -hypergeometric function, strong
See also:: Annotations for §17.12 and Ch.17

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

17.12.5	$\displaystyle\left(\frac{aq}{\rho_{1}},\frac{aq}{\rho_{2}};q\right)_{n}\alpha_% {n}^{\prime}$	$\displaystyle=\left(\rho_{1},\rho_{2};q\right)_{n}\left(\frac{aq}{\rho_{1}\rho% _{2}}\right)^{n}\alpha_{n}$
17.12.5	$\displaystyle\left(\frac{aq}{\rho_{1}},\frac{aq}{\rho_{2}};q\right)_{n}\beta_{% n}^{\prime}$	$\displaystyle=\sum_{j=0}^{n}\left(\rho_{1},\rho_{2};q\right)_{j}\left(\frac{aq% }{\rho_{1}\rho_{2}};q\right)_{n-j}\left(\frac{aq}{\rho_{1}\rho_{2}}\right)^{j}% \frac{\beta_{j}}{\left(q;q\right)_{n-j}}$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : 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 See also: Annotations for §17.12, §17.12 and Ch.17

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	$\displaystyle\alpha_{n}$	$\displaystyle=\frac{\left(a;q\right)_{n}(1-aq^{2n})(-1)^{n}q^{n(3n-1)/2}a^{n}}% {\left(q;q\right)_{n}(1-a)},$
17.12.6	$\displaystyle\beta_{n}$	$\displaystyle=\frac{1}{\left(q;q\right)_{n}}.$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (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 See also: Annotations for §17.12, §17.12 and Ch.17

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).

q

-Appell Functions 17.13 Integrals