17 q-Hypergeometric and Related Functions Properties 17.12 Bailey Pairs 17.14 Constant Term Identities

§17.13 Integrals

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Keywords:: integrals, $q$ -hypergeometric function
Notes:: See Gasper and Rahman (2004, p. 52), Berndt (1991, p. 29), Askey (1980).
Permalink:: http://dlmf.nist.gov/17.13
See also:: Annotations for Ch.17

In this section, for the function $\Gamma_{q}$ see §5.18(ii).

17.13.1		$\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-ax/c;q\right)_{\infty}\left(bx/d;q\right)_{\infty}}{\mathrm{d}}_{q}x=% \frac{(1-q)\left(q;q\right)_{\infty}\left(ab;q\right)_{\infty}cd\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(a;q\right)_{\infty}\left(b% ;q\right)_{\infty}(c+d)\left(-bc/d;q\right)_{\infty}\left(-ad/c;q\right)_{% \infty}},$
ⓘ Symbols: $\int$ : integral, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential, $q$ : complex base and $x$ : real variable Permalink: http://dlmf.nist.gov/17.13.E1 Encodings: TeX, pMML, png See also: Annotations for §17.13 and Ch.17

or, when $0<q<1$ ,

17.13.2		$\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-xq^{\alpha}/c;q\right)_{\infty}\left(xq^{\beta}/d;q\right)_{\infty}}{% \mathrm{d}}_{q}x=\frac{\Gamma_{q}\left(\alpha\right)\Gamma_{q}\left(\beta% \right)}{\Gamma_{q}\left(\alpha+\beta\right)}\frac{cd}{c+d}\frac{\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(-q^{\beta}c/d;q\right)_{% \infty}\left(-q^{\alpha}d/c;q\right)_{\infty}}.$
ⓘ Symbols: $\int$ : integral, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential, $q$ : complex base and $x$ : real variable Permalink: http://dlmf.nist.gov/17.13.E2 Encodings: TeX, pMML, png See also: Annotations for §17.13 and Ch.17

Ramanujan’s Integrals

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Keywords:: Ramanujan’s integrals, $q$ -hypergeometric function
See also:: Annotations for §17.13 and Ch.17

17.13.3		$\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{\alpha+\beta};q\right)_{\infty}}% {\left(-t;q\right)_{\infty}}\mathrm{d}t=\frac{\Gamma\left(\alpha\right)\Gamma% \left(1-\alpha\right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(1-\alpha% \right)\Gamma_{q}\left(\alpha+\beta\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial) and $q$ : complex base Referenced by: Erratum (V1.0.1) for Equation (17.13.3) Permalink: http://dlmf.nist.gov/17.13.E3 Encodings: TeX, pMML, png Errata (effective with 1.0.1): Originally the differential was identified incorrectly as a $q$ -differential; the correct differential is $\mathrm{d}t$ . See also: Annotations for §17.13, §17.13 and Ch.17

17.13.4		$\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-ctq^{\alpha+\beta};q\right)_{\infty}% }{\left(-ct;q\right)_{\infty}}{\mathrm{d}}_{q}t=\frac{\Gamma_{q}\left(\alpha% \right)\Gamma_{q}\left(\beta\right)\left(-cq^{\alpha};q\right)_{\infty}\left(-% q^{1-\alpha}/c;q\right)_{\infty}}{\Gamma_{q}\left(\alpha+\beta\right)\left(-c;% q\right)_{\infty}\left(-q/c;q\right)_{\infty}}.$
ⓘ Symbols: $\int$ : integral, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential and $q$ : complex base Permalink: http://dlmf.nist.gov/17.13.E4 Encodings: TeX, pMML, png See also: Annotations for §17.13, §17.13 and Ch.17

Askey (1980) conjectured extensions of the foregoing integrals that are closely related to Macdonald (1982). These conjectures are proved independently in Habsieger (1988) and Kadell (1988).