§17.13 Integrals

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

In this section, for the function $\mathop{\Gamma _{{q}}\/}\nolimits$ see §5.18(ii).

17.13.1 $\int _{{-c}}^{d}\frac{\left(-qx/c;q\right)_{{\infty}}\left(qx/d;q\right)_{{\infty}}}{\left(-ax/c;q\right)_{{\infty}}\left(bx/d;q\right)_{{\infty}}}{d}_{q}x=\frac{(1-q)\left(q;q\right)_{{\infty}}\left(ab;q\right)_{{\infty}}cd\left(-c/d;q\right)_{{\infty}}\left(-d/c;q\right)_{{\infty}}}{\left(a;q\right)_{{\infty}}\left(b;q\right)_{{\infty}}(c+d)\left(-bc/d;q\right)_{{\infty}}\left(-ad/c;q\right)_{{\infty}}},$

Symbols:: $\int$ : integral, ${d}_{q}x$ : $q$ -differential, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base and $x$ : real variable
Permalink:: http://dlmf.nist.gov/17.13.E1
Encodings:: TeX, pMML, png

or, when $0<q<1$ ,

17.13.2 $\int _{{-c}}^{d}\frac{\left(-qx/c;q\right)_{{\infty}}\left(qx/d;q\right)_{{\infty}}}{\left(-xq^{{\alpha}}/c;q\right)_{{\infty}}\left(xq^{{\beta}}/d;q\right)_{{\infty}}}{d}_{q}x=\frac{\mathop{\Gamma _{{q}}\/}\nolimits\!\left(\alpha\right)\mathop{\Gamma _{{q}}\/}\nolimits\!\left(\beta\right)}{\mathop{\Gamma _{{q}}\/}\nolimits\!\left(\alpha+\beta\right)}\frac{cd}{c+d}\frac{\left(-c/d;q\right)_{{\infty}}\left(-d/c;q\right)_{{\infty}}}{\left(-q^{{\beta}}c/d;q\right)_{{\infty}}\left(-q^{{\alpha}}d/c;q\right)_{{\infty}}}.$

Symbols:: $\int$ : integral, $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right)$ : $q$ -gamma function, ${d}_{q}x$ : $q$ -differential, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base and $x$ : real variable
Permalink:: http://dlmf.nist.gov/17.13.E2
Encodings:: TeX, pMML, png

¶ Ramanujan’s Integrals

Keywords:: $q$ -hypergeometric function

17.13.3 $\int _{0}^{\infty}t^{{\alpha-1}}\frac{\left(-tq^{{\alpha+\beta}};q\right)_{{\infty}}}{\left(-t;q\right)_{{\infty}}}dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(\alpha\right)\mathop{\Gamma\/}\nolimits\!\left(1-\alpha\right)\mathop{\Gamma _{{q}}\/}\nolimits\!\left(\beta\right)}{\mathop{\Gamma _{{q}}\/}\nolimits\!\left(1-\alpha\right)\mathop{\Gamma _{{q}}\/}\nolimits\!\left(\alpha+\beta\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right)$ : $q$ -gamma function, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial) and $q$ : complex base
Referenced by:: Version 1.0.1 (June 27, 2011)
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 $dt$ .
Reported 2011-04-08

17.13.4 $\int _{0}^{\infty}t^{{\alpha-1}}\frac{\left(-ctq^{{\alpha+\beta}};q\right)_{{\infty}}}{\left(-ct;q\right)_{{\infty}}}{d}_{q}t=\frac{\mathop{\Gamma _{{q}}\/}\nolimits\!\left(\alpha\right)\mathop{\Gamma _{{q}}\/}\nolimits\!\left(\beta\right)\left(-cq^{{\alpha}};q\right)_{{\infty}}\left(-q^{{1-\alpha}}/c;q\right)_{{\infty}}}{\mathop{\Gamma _{{q}}\/}\nolimits\!\left(\alpha+\beta\right)\left(-c;q\right)_{{\infty}}\left(-q/c;q\right)_{{\infty}}}.$

Symbols:: $\int$ : integral, $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right)$ : $q$ -gamma function, ${d}_{q}x$ : $q$ -differential, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial) and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.13.E4
Encodings:: TeX, pMML, png

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).