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

Referenced by:: §17.3(ii), Version 1.0.1 (June 27, 2011)
Permalink:: http://dlmf.nist.gov/5.18
Clarification (effective with 1.0.1):: The title was clarified to read $q$ -Beta, not Beta function.
Reported 2010-11-23

§5.18(i) $q$ -Factorials
§5.18(ii) $q$ -Gamma Function
§5.18(iii) $q$ -Beta Function

§5.18(i) $q$ -Factorials

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

5.18.1 $\left(a;q\right)_{{n}}=\prod _{{k=0}}^{{n-1}}(1-aq^{k}),$ $n=0,1,2,\dots$ ,

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (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

5.18.2 $n!_{q}=1(1+q)\cdots(1+q+\dots+q^{{n-1}})=\left(q;q\right)_{{n}}(1-q)^{{-n}}.$

Symbols:: $!$ : $n!$ : factorial, $\left(a;q\right)_{{n}}$ : $q$ -factorial (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

When $|q|<1$ ,

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (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

§5.18(ii) $q$ -Gamma Function

Notes:: See Andrews et al. (1999, pp. 485–495).
Keywords:: Bohr-Mollerup theorem, $q$ -gamma function
Referenced by:: §17.13
Permalink:: http://dlmf.nist.gov/5.18.ii

When $0<q<1$ ,

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:: $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right)$ : $q$ -gamma function
Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (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

5.18.5 $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(1\right)=\mathop{\Gamma _{{q}}\/}\nolimits\!\left(2\right)=1,$

Symbols:: $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right)$ : $q$ -gamma function and $q$ : real or complex variable
Permalink:: http://dlmf.nist.gov/5.18.E5
Encodings:: TeX, pMML, png

5.18.6 $n!_{q}=\mathop{\Gamma _{{q}}\/}\nolimits\!\left(n+1\right),$

Symbols:: $!$ : $n!$ : factorial, $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right)$ : $q$ -gamma function, $q$ : real or complex variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.18.E6
Encodings:: TeX, pMML, png

5.18.7 $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z+1\right)=\frac{1-q^{z}}{1-q}\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right)$ : $q$ -gamma function, $q$ : real or complex variable and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.18.E7
Encodings:: TeX, pMML, png

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 $0<q<r<1$ , then

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

Symbols:: $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right)$ : $q$ -gamma function, $q$ : real or complex variable and $x$ : real variable
Permalink:: http://dlmf.nist.gov/5.18.E8
Encodings:: TeX, pMML, png

when $0<x<1$ or when $x>2$ , and

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

Symbols:: $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right)$ : $q$ -gamma function, $q$ : real or complex variable and $x$ : real variable
Permalink:: http://dlmf.nist.gov/5.18.E9
Encodings:: TeX, pMML, png

when $1<x<2$ .

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right)$ : $q$ -gamma function, $q$ : real or complex variable and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.18.E10
Encodings:: TeX, pMML, png

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).

§5.18(iii) $q$ -Beta Function

Notes:: See Andrews et al. (1999, p. 494).
Keywords:: $q$ -beta function
Permalink:: http://dlmf.nist.gov/5.18.iii

5.18.11 $\mathop{\mathrm{B}_{{q}}\/}\nolimits\!\left(a,b\right)=\frac{\mathop{\Gamma _{{q}}\/}\nolimits\!\left(a\right)\mathop{\Gamma _{{q}}\/}\nolimits\!\left(b\right)}{\mathop{\Gamma _{{q}}\/}\nolimits\!\left(a+b\right)}.$

Defines:: $\mathop{\mathrm{B}_{{q}}\/}\nolimits\!\left(a,b\right)$ : $q$ -beta function
Symbols:: $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right)$ : $q$ -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

5.18.12 $\mathop{\mathrm{B}_{{q}}\/}\nolimits\!\left(a,b\right)=\int _{0}^{1}\frac{t^{{a-1}}\left(tq;q\right)_{{\infty}}}{\left(tq^{b};q\right)_{{\infty}}}{d}_{q}t,$ $0<q<1$ , $\realpart{a}>0$ , $\realpart{b}>0$ .

Symbols:: $\int$ : integral, $\mathop{\mathrm{B}_{{q}}\/}\nolimits\!\left(a,b\right)$ : $q$ -beta function, ${d}_{q}x$ : $q$ -differential, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\realpart{}$ : 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

For $q$ -integrals see §17.2(v).

§5.18 -Gamma and -Beta Functions

Contents

§5.18(i) -Factorials

§5.18(ii) -Gamma Function

§5.18(iii) -Beta Function

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

§5.18(i) $q$ -Factorials

§5.18(ii) $q$ -Gamma Function

§5.18(iii) $q$ -Beta Function