q-gamma function

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1: 5.18 $q$-Gamma and $q$-Beta Functions

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

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§5.18(ii) $q$-Gamma Function

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5.18.5 $${\mathrm{\Gamma }}_q\left(1\right)={\mathrm{\Gamma }}_q\left(2\right)=1,$$

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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:

pMML, png, TeX

See also:

Annotations for §5.18(ii), §5.18 and Ch.5

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

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Symbols:

${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${!}_q$: $q$-factorial (as in $n{!}_q$), $q$: real or complex variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.18.E6

Encodings:

pMML, png, TeX

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. …

2: 17.13 Integrals

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17.13.2 $${\int }_{-c}^d\frac{{(-qx/c;q)}_{\mathrm{\infty }}{(qx/d;q)}_{\mathrm{\infty }}}{{(-x{q}^{\alpha }/c;q)}_{\mathrm{\infty }}{(x{q}^{\beta }/d;q)}_{\mathrm{\infty }}}{d}_{q}x=\frac{{\mathrm{\Gamma }}_q\left(\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(\alpha +\beta \right)}\frac{cd}{c+d}\frac{{(-c/d;q)}_{\mathrm{\infty }}{(-d/c;q)}_{\mathrm{\infty }}}{{(-q^{\beta }c/d;q)}_{\mathrm{\infty }}{(-q^{\alpha }d/c;q)}_{\mathrm{\infty }}}.$$

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Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13 and Ch.17

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17.13.3 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-t{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-t;q)}_{\mathrm{\infty }}}dt=\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\alpha +\beta \right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial) and $q$: complex base

Referenced by:

Erratum (V1.0.1) for Equation (17.13.3)

Permalink:

http://dlmf.nist.gov/17.13.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally the differential was identified incorrectly as a $q$-differential; the correct differential is $dt$.

See also:

Annotations for §17.13, §17.13 and Ch.17

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17.13.4 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-ct{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-ct;q)}_{\mathrm{\infty }}}{d}_{q}t=\frac{{\mathrm{\Gamma }}_q\left(\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right){(-c{q}^{\alpha };q)}_{\mathrm{\infty }}{(-q^{1-\alpha }/c;q)}_{\mathrm{\infty }}}{{\mathrm{\Gamma }}_q\left(\alpha +\beta \right){(-c;q)}_{\mathrm{\infty }}{(-q/c;q)}_{\mathrm{\infty }}}.$$

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Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13, §17.13 and Ch.17

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3: 5.1 Special Notation

… ►The main functions treated in this chapter are the gamma function $\mathrm{\Gamma }\left(z\right)$, the psi function (or digamma function) $\psi \left(z\right)$, the beta function $\mathrm{B}(a,b)$, and the $q$-gamma function ${\mathrm{\Gamma }}_q\left(z\right)$. …

4: 5.21 Methods of Computation

… ►For the computation of the $q$-gamma and $q$-beta functions see Gabutti and Allasia (2008).

5: Bibliography G

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Evaluation of $q$-gamma function and $q$-analogues by iterative algorithms. Numer. Algorithms 49 (1-4), pp. 159–168.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview (Necdet Batir), ZentralBlatt

Referenced by:

§17.18, §17.18, §5.21, §5.21.

Encodings:

BibTeX

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6: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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17.6.28 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{\alpha },q^{\beta }}{q^{\gamma }};q,z)=\frac{{\mathrm{\Gamma }}_q\left(\gamma \right)}{{\mathrm{\Gamma }}_q\left(\beta \right){\mathrm{\Gamma }}_q\left(\gamma -\beta \right)}{\int }_0^1\frac{t^{\beta -1}{(tq;q)}_{\gamma -\beta -1}}{{(xt;q)}_{\alpha }}{d}_{q}t.$$

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Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $z$: complex variable and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.6.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §17.6(v), §17.6 and Ch.17

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7: Bibliography O

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Asymptotic expansions for $q$-gamma, $q$-exponential, and $q$-Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.

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External Links:

ISSN 0022-247X, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§5.18(ii).

Encodings:

BibTeX

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