5 Gamma Function Properties 5.17 Barnes’

G

-Function (Double Gamma Function)5.19 Mathematical Applications

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

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Referenced by:: §17.3(ii), Erratum (V1.0.1) for Clarifications
Permalink:: http://dlmf.nist.gov/5.18
Clarification (effective with 1.0.1):: The title was clarified to read $q$ -Beta, not Beta function.
See also:: Annotations for Ch.5

§5.18(i) $q$ -Factorials

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Keywords:: $q$ -factorials
Notes:: See Andrews et al. (1999, pp. 487–488).
Permalink:: http://dlmf.nist.gov/5.18.i
See also:: Annotations for §5.18 and Ch.5

5.18.1		$\left(a;q\right)_{n}=\prod_{k=0}^{n-1}(1-aq^{k}),$
$n=0,1,2,\dots$ ,
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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		$n!_{q}=1(1+q)\cdots(1+q+\dots+q^{n-1})=\left(q;q\right)_{n}(1-q)^{-n}.$
ⓘ Defines: $!_{\NVar{q}}$ : $q$ -factorial (as in $n!_{q}$ ) Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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), §5.18 and Ch.5

When $|q|<1$ ,

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

§5.18(ii) $q$ -Gamma Function

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Keywords:: Bohr-Mollerup theorem, $q$ -gamma function
Notes:: See Andrews et al. (1999, pp. 485–495).
Referenced by:: §17.13, Erratum (V1.0.10) for References
Permalink:: http://dlmf.nist.gov/5.18.ii
Addition (effective with 1.0.10):: A sentence and reference to Salem (2013) was added at the end of this subsection.
See also:: Annotations for §5.18 and Ch.5

When $0<q<1$ ,

5.18.4		$\Gamma_{q}\left(z\right)=\left(q;q\right)_{\infty}(1-q)^{1-z}/\left(q^{z};q% \right)_{\infty},$
ⓘ Defines: $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -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 and Ch.5

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

5.18.6		$n!_{q}=\Gamma_{q}\left(n+1\right),$
ⓘ Symbols: $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $!_{\NVar{q}}$ : $q$ -factorial (as in $n!_{q}$ ), $q$ : 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		$\Gamma_{q}\left(z+1\right)=\frac{1-q^{z}}{1-q}\Gamma_{q}\left(z\right).$
ⓘ Symbols: $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -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, $\ln\Gamma_{q}\left(x\right)$ is convex for $x>0$ , and the analog of the Bohr–Mollerup theorem (§5.5(iv)) holds.

If $0<q<r<1$ , then

5.18.8		$\Gamma_{q}\left(x\right)<\Gamma_{r}\left(x\right),$
ⓘ Symbols: $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -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 $0<x<1$ or when $x>2$ , and

5.18.9		$\Gamma_{q}\left(x\right)>\Gamma_{r}\left(x\right),$
ⓘ Symbols: $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -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 $1<x<2$ .

5.18.10		$\lim_{q\to 1-}\Gamma_{q}\left(z\right)=\Gamma\left(z\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -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 $\ln\Gamma_{q}\left(z\right)$ as $|z|\to\infty$ see Olde Daalhuis (1994) and Moak (1984). For the $q$ -digamma or $q$ -psi function $\psi_{q}(z)=\Gamma_{q}'\left(z\right)/\Gamma_{q}\left(z\right)$ see Salem (2013).

§5.18(iii) $q$ -Beta Function

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Keywords:: $q$ -beta function
Notes:: See Andrews et al. (1999, p. 494).
Permalink:: http://dlmf.nist.gov/5.18.iii
See also:: Annotations for §5.18 and Ch.5

5.18.11	$\displaystyle\mathrm{B}_{q}\left(a,b\right)$	$\displaystyle=\frac{\Gamma_{q}\left(a\right)\Gamma_{q}\left(b\right)}{\Gamma_{% q}\left(a+b\right)}.$
ⓘ Defines: $\mathrm{B}_{\NVar{q}}\left(\NVar{a},\NVar{b}\right)$ : $q$ -beta function Symbols: $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -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 and Ch.5
5.18.12	$\displaystyle\mathrm{B}_{q}\left(a,b\right)$	$\displaystyle=\int_{0}^{1}\frac{t^{a-1}\left(tq;q\right)_{\infty}}{\left(tq^{b% };q\right)_{\infty}}\,{\mathrm{d}}_{q}t,$
$0<q<1$ , $\Re a>0$ , $\Re b>0$ .
ⓘ Symbols: $\int$ : integral, $\mathrm{B}_{\NVar{q}}\left(\NVar{a},\NVar{b}\right)$ : $q$ -beta function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential, $\Re$ : 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