q-gamma function

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1: 5.18 $q$-Gamma and $q$-Beta Functions
§5.18(ii) $q$-GammaFunction
5.18.5 $\Gamma_{q}\left(1\right)=\Gamma_{q}\left(2\right)=1,$
5.18.6 $n!_{q}=\Gamma_{q}\left(n+1\right),$
Also, $\ln\Gamma_{q}\left(x\right)$ is convex for $x>0$, and the analog of the Bohr–Mollerup theorem (§5.5(iv)) holds. …
2: 17.13 Integrals
17.13.2 $\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-xq^{\alpha}/c;q\right)_{\infty}\left(xq^{\beta}/d;q\right)_{\infty}}\,{% \mathrm{d}}_{q}x=\frac{\Gamma_{q}\left(\alpha\right)\Gamma_{q}\left(\beta% \right)}{\Gamma_{q}\left(\alpha+\beta\right)}\frac{cd}{c+d}\frac{\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(-q^{\beta}c/d;q\right)_{% \infty}\left(-q^{\alpha}d/c;q\right)_{\infty}}.$
17.13.3 $\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{\alpha+\beta};q\right)_{\infty}}% {\left(-t;q\right)_{\infty}}\,\mathrm{d}t=\frac{\Gamma\left(\alpha\right)% \Gamma\left(1-\alpha\right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(1-% \alpha\right)\Gamma_{q}\left(\alpha+\beta\right)},$
17.13.4 $\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-ctq^{\alpha+\beta};q\right)_{\infty}% }{\left(-ct;q\right)_{\infty}}\,{\mathrm{d}}_{q}t=\frac{\Gamma_{q}\left(\alpha% \right)\Gamma_{q}\left(\beta\right)\left(-cq^{\alpha};q\right)_{\infty}\left(-% q^{1-\alpha}/c;q\right)_{\infty}}{\Gamma_{q}\left(\alpha+\beta\right)\left(-c;% q\right)_{\infty}\left(-q/c;q\right)_{\infty}}.$
3: 5.1 Special Notation
The main functions treated in this chapter are the gamma function $\Gamma\left(z\right)$, the psi function (or digamma function) $\psi\left(z\right)$, the beta function $\mathrm{B}\left(a,b\right)$, and the $q$-gamma function $\Gamma_{q}\left(z\right)$. …
4: 5.21 Methods of Computation
For the computation of the $q$-gamma and $q$-beta functions see Gabutti and Allasia (2008).
5: Bibliography G
• B. Gabutti and G. Allasia (2008) Evaluation of $q$-gamma function and $q$-analogues by iterative algorithms. Numer. Algorithms 49 (1-4), pp. 159–168.
• 6: 17.6 ${{}_{2}\phi_{1}}$ Function
17.6.28 ${{}_{2}\phi_{1}}\left({q^{\alpha},q^{\beta}\atop q^{\gamma}};q,z\right)=\frac{% \Gamma_{q}\left(\gamma\right)}{\Gamma_{q}\left(\beta\right)\Gamma_{q}\left(% \gamma-\beta\right)}\int_{0}^{1}\frac{t^{\beta-1}\left(tq;q\right)_{\gamma-% \beta-1}}{\left(xt;q\right)_{\alpha}}\,{\mathrm{d}}_{q}t.$
7: Bibliography O
• A. B. Olde Daalhuis (1994) Asymptotic expansions for $q$-gamma, $q$-exponential, and $q$-Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.