§ 5.18. $q$ -Gamma and Beta Functions

Permalink:: http://dlmf.nist.gov/5.18

§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).
Permalink:: http://dlmf.nist.gov/5.18.SS1

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

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

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

Defines:: $\left(a;q\right)_{{n}}$ : $q$ -factorial
Symbols:: $q$ : real or complex variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.18.E2
Encodings:: TeX, pMathML, png

When $|q|<1$ ,

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

Defines:: $\left(a;q\right)_{{n}}$ : $q$ -factorial
Symbols:: $q$ : real or complex variable, $k$ : nonnegative integer and $a$ : real or complex variable
Permalink:: http://dlmf.nist.gov/5.18.E3
Encodings:: TeX, pMathML, png

§ 5.18(ii). $q$ -Gamma Function

Notes:: See Andrews et al. (1999, pp. 485–495).
Permalink:: http://dlmf.nist.gov/5.18.SS2

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:: $\Gamma _{{q}}\!\left(z\right)$ : $q$ -Gamma function
Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial, $q$ : real or complex variable and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.18.E4
Encodings:: TeX, pMathML, png

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

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

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

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

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

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

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

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

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

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

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

when $1<x<2$ .

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

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

For generalized asymptotic expansions of $\ln\Gamma _{{q}}\!\left(z\right)$ 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).
Permalink:: http://dlmf.nist.gov/5.18.SS3

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

Defines:: $\mathrm{B}_{{q}}\!\left(a,b\right)$ : $q$ -Beta function
Symbols:: $\Gamma _{{q}}\!\left(z\right)$ : $q$ -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, pMathML, png

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

Defines:: $\mathrm{B}_{{q}}\!\left(a,b\right)$ : $q$ -Beta function
Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial, $q$ : 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, pMathML, png

For $q$ -integrals see §Ch.17.

§ 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 Beta Functions

§ 5.18(i). $q$ -Factorials

§ 5.18(ii). $q$ -Gamma Function

§ 5.18(iii). $q$ -Beta Function