DLMF:17.13.E3

MaRDI QIDQ6165

• Label: DLMF:17.13.E3

Digital Library of Mathematical Functions ID 17.13.E3

${\displaystyle{\displaystyle\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{\alpha+\beta};q\right)_{\infty}}{\left(-t;q\right)_{\infty}}\mathrm{d}t=\frac{\Gamma\left(\alpha\right)\Gamma\left(1-\alpha\right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(1-\alpha\right)\Gamma_{q}\left(\alpha+\beta\right)},}}$

Symbols List

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

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