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

§17.13 Integrals

Notes:: See Gasper and Rahman (2004, p. 52), Berndt (1991, p. 29), Askey (1980).
Keywords:: integrals, -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, ${d}_{q}x$ : -differential, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base and : real variable
Permalink:: http://dlmf.nist.gov/17.13.E1
Encodings:: TeX, pMML, png

or, when ,

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

¶ Ramanujan’s Integrals

Keywords:: -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, : differential of , $\int$ : integral, $\mathop{\Gamma_{{q}}\/}\nolimits\!\left(z\right)$ : -gamma function, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial) and : complex base
Referenced by:: 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 -differential; the correct differential is .
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, $\mathop{\Gamma_{{q}}\/}\nolimits\!\left(z\right)$ : -gamma function, ${d}_{q}x$ : -differential, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial) and : 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).

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