# §5.18 $q$-Gamma and $q$-Beta Functions

## §5.18(i) $q$-Factorials

 5.18.1 $\left(a;q\right)_{n}=\prod_{k=0}^{n-1}(1-aq^{k}),$ $n=0,1,2,\dots$,
 5.18.2 $n!_{q}=1(1+q)\cdots(1+q+\dots+q^{n-1})=\left(q;q\right)_{n}(1-q)^{-n}.$ Defines: $!_{\NVar{q}}$: $q$-factorial (as in $n!_{q}$) Symbols: $!$: factorial (as in $n!$), $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/5.18.E2 Encodings: TeX, pMML, png See also: Annotations for 5.18(i)

When $|q|<1$,

 5.18.3 $\left(a;q\right)_{\infty}=\prod_{k=0}^{\infty}(1-aq^{k}).$

## §5.18(ii) $q$-Gamma Function

When $0,

 5.18.4 $\mathop{\Gamma_{q}\/}\nolimits\!\left(z\right)=\left(q;q\right)_{\infty}(1-q)^% {1-z}/\left(q^{z};q\right)_{\infty},$ Defines: $\mathop{\Gamma_{\NVar{q}}\/}\nolimits\!\left(\NVar{z}\right)$: $q$-gamma function Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable and $z$: complex variable Permalink: http://dlmf.nist.gov/5.18.E4 Encodings: TeX, pMML, png See also: Annotations for 5.18(ii)
 5.18.5 $\mathop{\Gamma_{q}\/}\nolimits\!\left(1\right)=\mathop{\Gamma_{q}\/}\nolimits% \!\left(2\right)=1,$ Symbols: $\mathop{\Gamma_{\NVar{q}}\/}\nolimits\!\left(\NVar{z}\right)$: $q$-gamma function and $q$: real or complex variable Permalink: http://dlmf.nist.gov/5.18.E5 Encodings: TeX, pMML, png See also: Annotations for 5.18(ii)
 5.18.6 $n!_{q}=\mathop{\Gamma_{q}\/}\nolimits\!\left(n+1\right),$
 5.18.7 $\mathop{\Gamma_{q}\/}\nolimits\!\left(z+1\right)=\frac{1-q^{z}}{1-q}\mathop{% \Gamma_{q}\/}\nolimits\!\left(z\right).$

Also, $\mathop{\ln\/}\nolimits\mathop{\Gamma_{q}\/}\nolimits\!\left(x\right)$ is convex for $x>0$, and the analog of the Bohr-Mollerup theorem (§5.5(iv)) holds.

If $0, then

 5.18.8 $\mathop{\Gamma_{q}\/}\nolimits\!\left(x\right)<\mathop{\Gamma_{r}\/}\nolimits% \!\left(x\right),$

when $0 or when $x>2$, and

 5.18.9 $\mathop{\Gamma_{q}\/}\nolimits\!\left(x\right)>\mathop{\Gamma_{r}\/}\nolimits% \!\left(x\right),$

when $1.

 5.18.10 $\lim_{q\to 1-}\mathop{\Gamma_{q}\/}\nolimits\!\left(z\right)=\mathop{\Gamma\/}% \nolimits\!\left(z\right).$

For generalized asymptotic expansions of $\mathop{\ln\/}\nolimits\mathop{\Gamma_{q}\/}\nolimits\!\left(z\right)$ as $|z|\to\infty$ see Olde Daalhuis (1994) and Moak (1984). For the $q$-digamma or $q$-psi function $\psi_{q}(z)=\mathop{\Gamma_{q}\/}\nolimits'\!\left(z\right)/\mathop{\Gamma_{q}% \/}\nolimits\!\left(z\right)$ see Salem (2013).

## §5.18(iii) $q$-Beta Function

 5.18.11 $\displaystyle\mathop{\mathrm{B}_{q}\/}\nolimits\!\left(a,b\right)$ $\displaystyle=\frac{\mathop{\Gamma_{q}\/}\nolimits\!\left(a\right)\mathop{% \Gamma_{q}\/}\nolimits\!\left(b\right)}{\mathop{\Gamma_{q}\/}\nolimits\!\left(% a+b\right)}.$ Defines: $\mathop{\mathrm{B}_{\NVar{q}}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: $q$-beta function Symbols: $\mathop{\Gamma_{\NVar{q}}\/}\nolimits\!\left(\NVar{z}\right)$: $q$-gamma function, $q$: real or complex variable, $a$: real or complex variable and $b$: real or complex variable Permalink: http://dlmf.nist.gov/5.18.E11 Encodings: TeX, pMML, png See also: Annotations for 5.18(iii) 5.18.12 $\displaystyle\mathop{\mathrm{B}_{q}\/}\nolimits\!\left(a,b\right)$ $\displaystyle=\int_{0}^{1}\frac{t^{a-1}\left(tq;q\right)_{\infty}}{\left(tq^{b% };q\right)_{\infty}}{\mathrm{d}}_{q}t,$ $0, $\Re{a}>0$, $\Re{b}>0$.

For $q$-integrals see §17.2(v).