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

5.18.1		$${(a;q)}_n=\prod _{k=0}^{n-1}(1-a{q}^k),$$
$n=0,1,2,\mathrm{\dots }$,
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable, $n$: nonnegative integer, $k$: nonnegative integer and $a$: real or complex variable Permalink: http://dlmf.nist.gov/5.18.E1 Encodings: TeX, pMML, png See also: Annotations for §5.18(i), §5.18 and Ch.5

5.18.2		$$n{\mathit{!}}_q=1(1+q)\mathrm{\cdots }(1+q+\mathrm{\cdots }+q^{n-1})={(q;q)}_n{(1-q)}^{-n}.$$
ⓘ Defines: ${\mathit{!}}_q$: $q$-factorial (as in $n{\mathit{!}}_q$) Symbols: ${(a;q)}_n$: $q$-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), §5.18 and Ch.5

When ,

5.18.3		$${(a;q)}_{\mathrm{\infty }}=\prod _{k=0}^{\mathrm{\infty }}(1-a{q}^k).$$
ⓘ Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable, $k$: nonnegative integer and $a$: real or complex variable Permalink: http://dlmf.nist.gov/5.18.E3 Encodings: TeX, pMML, png See also: Annotations for §5.18(i), §5.18 and Ch.5

See also §17.2(i).

When ,

5.18.4		$${\mathrm{\Gamma }}_q\left(z\right)={(q;q)}_{\mathrm{\infty }}{(1-q)}^{1-z}/{(q^z;q)}_{\mathrm{\infty }},$$
ⓘ Defines: ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function Symbols: ${(a;q)}_n$: $q$-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 and Ch.5

5.18.5		$${\mathrm{\Gamma }}_q\left(1\right)={\mathrm{\Gamma }}_q\left(2\right)=1,$$
ⓘ Symbols: ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-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 and Ch.5

5.18.6		$$n{\mathit{!}}_q={\mathrm{\Gamma }}_q\left(n+1\right),$$
ⓘ Symbols: ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${\mathit{!}}_q$: $q$-factorial (as in $n{\mathit{!}}_q$), $q$: real or complex variable and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/5.18.E6 Encodings: TeX, pMML, png See also: Annotations for §5.18(ii), §5.18 and Ch.5

5.18.7		$${\mathrm{\Gamma }}_q\left(z+1\right)=\frac{1-q^z}{1-q}{\mathrm{\Gamma }}_q\left(z\right).$$
ⓘ Symbols: ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, $q$: real or complex variable and $z$: complex variable Permalink: http://dlmf.nist.gov/5.18.E7 Encodings: TeX, pMML, png See also: Annotations for §5.18(ii), §5.18 and Ch.5

Also, $\ln {\mathrm{\Gamma }}_q\left(x\right)$ is convex for $x>0$, and the analog of the Bohr-Mollerup theorem (§5.5(iv)) holds.

If , then

5.18.8
ⓘ Symbols: ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, $q$: real or complex variable and $x$: real variable Permalink: http://dlmf.nist.gov/5.18.E8 Encodings: TeX, pMML, png See also: Annotations for §5.18(ii), §5.18 and Ch.5

when or when $x>2$, and

5.18.9		$${\mathrm{\Gamma }}_q\left(x\right)>{\mathrm{\Gamma }}_r\left(x\right),$$
ⓘ Symbols: ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, $q$: real or complex variable and $x$: real variable Permalink: http://dlmf.nist.gov/5.18.E9 Encodings: TeX, pMML, png See also: Annotations for §5.18(ii), §5.18 and Ch.5

when .

5.18.10		$$\underset{q\to 1-}{lim}{\mathrm{\Gamma }}_q\left(z\right)=\mathrm{\Gamma }\left(z\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, $q$: real or complex variable and $z$: complex variable Permalink: http://dlmf.nist.gov/5.18.E10 Encodings: TeX, pMML, png See also: Annotations for §5.18(ii), §5.18 and Ch.5

For generalized asymptotic expansions of $\ln {\mathrm{\Gamma }}_q\left(z\right)$ as $|z|\to \mathrm{\infty }$ see Olde Daalhuis (1994) and Moak (1984). For the $q$-digamma or $q$-psi function ${\psi }_q(z)={\mathrm{\Gamma }}_q^{\prime }\left(z\right)/{\mathrm{\Gamma }}_q\left(z\right)$ see Salem (2013).

5.18.11		${\mathrm{B}}_q(a,b)$	$={\displaystyle \frac{{\mathrm{\Gamma }}_q\left(a\right){\mathrm{\Gamma }}_q\left(b\right)}{{\mathrm{\Gamma }}_q\left(a+b\right)}}.$
ⓘ Defines: ${\mathrm{B}}_q(a,b)$: $q$-beta function Symbols: ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-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 and Ch.5
5.18.12		${\mathrm{B}}_q(a,b)$	$={\displaystyle {\int }_0^1}{\displaystyle \frac{t^{a-1}{(tq;q)}_{\mathrm{\infty }}}{{(t{q}^b;q)}_{\mathrm{\infty }}}}{d}_{q}t,$
, $\mathrm{\Re }a>0$, $\mathrm{\Re }b>0$.
ⓘ Symbols: $\int$: integral, ${\mathrm{B}}_q(a,b)$: $q$-beta function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $\mathrm{\Re }$: real part, $q$: real or complex variable, $a$: real or complex variable and $b$: real or complex variable Permalink: http://dlmf.nist.gov/5.18.E12 Encodings: TeX, pMML, png See also: Annotations for §5.18(iii), §5.18 and Ch.5

For $q$-integrals see §17.2(v).