Formula:6092

Digital Library of Mathematical Functions ID 17.6.E28

${\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({q^{\alpha},q^{\beta}\atop q^{\gamma}};q,z\right)=\frac{\Gamma_{q}\left(\gamma\right)}{\Gamma_{q}\left(\beta\right)\Gamma_{q}\left(\gamma-\beta\right)}\int_{0}^{1}\frac{t^{\beta-1}\left(tq;q\right)_{\gamma-\beta-1}}{\left(xt;q\right)_{\alpha}}{\mathrm{d}}_{q}t.}}$

Symbols List

• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Gamma_{\NVar{q}}\left(\NVar{z}\right)}}$: $$q$$ -gamma function
• ${\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle{\mathrm{d}}_{\NVar{q}}\NVar{x}}}$:
• ${\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}$: basic hypergeometric (or $$q$$ -hypergeometric) function
• ${\displaystyle{\displaystyle q}}$: complex base
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle x}}$: real variable
 

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