5 Gamma Function Properties 5.17 Barnes’ $\mathop{G\/}\nolimits$ -Function (Double Gamma Function)5.19 Mathematical Applications

§5.18 -Gamma and -Beta Functions

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

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

§5.18(i) -Factorials

Notes:: See Andrews et al. (1999, pp. 487–488).
Keywords:: -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:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real or complex variable, : nonnegative integer, : nonnegative integer and : 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:: : : factorial, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real or complex variable and : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.18.E2
Encodings:: TeX, pMML, png

When ,

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

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

§5.18(ii) -Gamma Function

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

When ,

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:: $\mathop{\Gamma_{{q}}\/}\nolimits\!\left(z\right)$ : -gamma function
Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : real or complex variable and : 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:: $\mathop{\Gamma_{{q}}\/}\nolimits\!\left(z\right)$ : -gamma function and : 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:: : : factorial, $\mathop{\Gamma_{{q}}\/}\nolimits\!\left(z\right)$ : -gamma function, : real or complex variable and : 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:: $\mathop{\Gamma_{{q}}\/}\nolimits\!\left(z\right)$ : -gamma function, : real or complex variable and : 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 , and the analog of the Bohr-Mollerup theorem (§5.5(iv)) holds.

If , then

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

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

when or when , and

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

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

when .

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, $\mathop{\Gamma_{{q}}\/}\nolimits\!\left(z\right)$ : -gamma function, : real or complex variable and : 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) -Beta Function

Notes:: See Andrews et al. (1999, p. 494).
Keywords:: -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:: $\mathop{\mathrm{B}_{{q}}\/}\nolimits\!\left(a,b\right)$ : -beta function
Symbols:: $\mathop{\Gamma_{{q}}\/}\nolimits\!\left(z\right)$ : -gamma function, : real or complex variable, : real or complex variable and : real or complex variable
Permalink:: http://dlmf.nist.gov/5.18.E11
Encodings:: TeX, pMML, png

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

, $\realpart{a}>0$ , $\realpart{b}>0$ .

Symbols:: $\int$ : integral, $\mathop{\mathrm{B}_{{q}}\/}\nolimits\!\left(a,b\right)$ : -beta function, ${d}_{q}x$ : -differential, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), $\realpart{}$ : real part, : real or complex variable, : real or complex variable and : real or complex variable
Permalink:: http://dlmf.nist.gov/5.18.E12
Encodings:: TeX, pMML, png

For -integrals see §17.2(v).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 5.17 Barnes’ $\mathop{G\/}\nolimits$ -Function (Double Gamma Function)5.19 Mathematical Applications

§5.18 -Gamma and -Beta Functions

Contents

§5.18(i) -Factorials

§5.18(ii) -Gamma Function

§5.18(iii) -Beta Function