Formula:6166

Digital Library of Mathematical Functions ID 17.13.E4

${\displaystyle{\displaystyle\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-ctq^{\alpha+\beta};q\right)_{\infty}}{\left(-ct;q\right)_{\infty}}{\mathrm{d}}_{q}t=\frac{\Gamma_{q}\left(\alpha\right)\Gamma_{q}\left(\beta\right)\left(-cq^{\alpha};q\right)_{\infty}\left(-q^{1-\alpha}/c;q\right)_{\infty}}{\Gamma_{q}\left(\alpha+\beta\right)\left(-c;q\right)_{\infty}\left(-q/c;q\right)_{\infty}}.}}$

Symbols List

• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Gamma_{\NVar{q}}\left(\NVar{z}\right)}}$: $$q$$ -gamma function
• ${\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle{\mathrm{d}}_{\NVar{q}}\NVar{x}}}$:
• ${\displaystyle{\displaystyle q}}$: complex base
 

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