q-gamma function

In q-analog theory, the $q$ -gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by {{harvtxt|Jackson|1905}}. It is given by

$\Gamma_q(x) = (1-q)^{1-x}\prod_{n=0}^\infty \frac{1-q^{n+1}}{1-q^{n+x}}=(1-q)^{1-x}\,\frac{(q;q)_\infty}{(q^x;q)_\infty}$

when $|q|<1$ , and

$\Gamma_q(x)=\frac{(q^{-1};q^{-1})_\infty}{(q^{-x};q^{-1})_\infty}(q-1)^{1-x}q^{\binom{x}{2}}$

if $|q|>1$ . Here $(\cdot;\cdot)_\infty$ is the infinite $q$ -Pochhammer symbol. The $q$ -gamma function satisfies the functional equation

$\Gamma_q(x+1) = \frac{1-q^{x}}{1-q}\Gamma_q(x)=[x]_q\Gamma_q(x)$

In addition, the $q$ -gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey ({{harvtxt|Askey|1978}}).

For non-negative integers $n$ ,

$\Gamma_q(n)=[n-1]_q!$

where $[\cdot]_q$ is the $q$ -factorial function. Thus the $q$ -gamma function can be considered as an extension of the $q$ -factorial function to the real numbers.

The relation to the ordinary gamma function is made explicit in the limit

$\lim_{q \to 1\pm} \Gamma_q(x) = \Gamma(x).$

There is a simple proof of this limit by Gosper. See the appendix of ({{harvs|txt|authorlink=George Andrews (mathematician)| last=Andrews| year=1986}}).

Transformation properties

The $q$ -gamma function satisfies the q-analog of the Gauss multiplication formula ({{harvtxt|Gasper|Rahman|2004}}):

$\Gamma_q(nx)\Gamma_r(1/n)\Gamma_r(2/n)\cdots\Gamma_r((n-1)/n)=\left(\frac{1-q^n}{1-q}\right)^{nx-1}\Gamma_r(x)\Gamma_r(x+1/n)\cdots\Gamma_r(x+(n-1)/n), \ r=q^n.$

= Integral representation =

The $q$ -gamma function has the following integral representation ({{harvs|txt|authorlink=Mourad E. H. Ismail| last=Ismail| year=1981}}):

$\frac{1}{\Gamma_q(z)}=\frac{\sin(\pi z)}{\pi}\int_0^\infty\frac{t^{-z}\mathrm{d}t}{(-t(1-q);q)_{\infty}}.$

= Stirling formula =

Moak obtained the following q-analogue of the Stirling formula (see {{harvtxt|Moak|1984}}):

$\log\Gamma_q(x)\sim(x-1/2)\log[x]_q+\frac{\mathrm{Li}_2(1-q^x)}{\log q}+C_{\hat{q}}+\frac{1}{2}H(q-1)\log q+\sum_{k=1}^\infty
\frac{B_{2k}}{(2k)!}\left(\frac{\log \hat{q}}{\hat{q}^x-1}\right)^{2k-1}\hat{q}^x p_{2k-3}(\hat{q}^x), \ x\to\infty,$

$\hat{q}=
\left\{\begin{aligned}
q \quad \mathrm{if} \ &0$

$C_q = \frac{1}{2} \log(2\pi)+\frac{1}{2}\log\left(\frac{q-1}{\log q}\right)-\frac{1}{24}\log q+\log\sum_{m=-\infty}^\infty \left(r^{m(6m+1)} - r^{(3m+1)(2m+1)}\right),$

where $r=\exp(4\pi^2/\log q)$ , $H$ denotes the Heaviside step function, $B_k$ stands for the Bernoulli number, $\mathrm{Li}_2(z)$ is the dilogarithm, and $p_k$ is a polynomial of degree $k$ satisfying

$p_k(z)=z(1-z)p'_{k-1}(z)+(kz+1)p_{k-1}(z), p_0=p_{-1}=1, k=1,2,\cdots.$

Raabe-type formulas

Due to I. Mező, the q-analogue of the Raabe formula exists, at least if we use the $q$ -gamma function when $|q|>1$ . With this restriction,

$\int_0^1\log\Gamma_q(x)dx=\frac{\zeta(2)}{\log q}+\log\sqrt{\frac{q-1}{\sqrt[6]{q}}}+\log(q^{-1};q^{-1})_\infty \quad(q>1).$

El Bachraoui considered the case $0 and proved that$

$\int_0^1\log\Gamma_q(x)dx=\frac{1}{2}\log (1-q) - \frac{\zeta(2)}{\log q}+\log(q;q)_\infty \quad(0$

Special values

The following special values are known.{{cite arXiv |last=Mező |first=István |year=2011 |title=Several special values of Jacobi theta functions |eprint=1106.1042 |mode=cs2 |class=math.NT }}

$\Gamma_{e^{-\pi}}\left(\frac12\right)=\frac{e^{-7 \pi /16} \sqrt{e^\pi-1}\sqrt[4]{1+\sqrt2}}{2^{15/16}\pi^{3/4}} \, \Gamma \left(\frac{1}{4}\right),$

$\Gamma_{e^{-2\pi}}\left(\frac12\right)=\frac{e^{-7 \pi /8} \sqrt{e^{2 \pi}-1}}{2^{9/8} \pi^{3/4}} \, \Gamma \left(\frac{1}{4}\right),$

$\Gamma_{e^{-4\pi}}\left(\frac12\right)=\frac{e^{-7 \pi /4} \sqrt{e^{4 \pi}-1}}{2^{7/4} \pi^{3/4}} \, \Gamma \left(\frac{1}{4}\right),$

$\Gamma_{e^{-8\pi}}\left(\frac12\right)=\frac{e^{-7 \pi /2} \sqrt{e^{8 \pi}-1}}{2^{9/4} \pi^{3/4} \sqrt{1+\sqrt2}} \, \Gamma \left(\frac{1}{4}\right).$

These are the analogues of the classical formula $\Gamma\left(\frac12\right)=\sqrt\pi$ .

Moreover, the following analogues of the familiar identity $\Gamma\left(\frac14\right)\Gamma\left(\frac34\right)=\sqrt2\pi$ hold true:

$\Gamma_{e^{-2\pi}}\left(\frac14\right)\Gamma_{e^{-2\pi}}\left(\frac34\right)=\frac{e^{-29 \pi /16} \left(e^{2 \pi }-1\right)\sqrt[4]{1+\sqrt2}}{2^{33/16} \pi^{3/2}} \, \Gamma \left(\frac{1}{4}\right)^2,$

$\Gamma_{e^{-4\pi}}\left(\frac14\right)\Gamma_{e^{-4\pi}}\left(\frac34\right)=\frac{e^{-29 \pi /8} \left(e^{4 \pi }-1\right)}{2^{23/8} \pi ^{3/2}} \, \Gamma \left(\frac{1}{4}\right)^2,$

$\Gamma_{e^{-8\pi}}\left(\frac14\right)\Gamma_{e^{-8\pi}}\left(\frac34\right)=\frac{e^{-29 \pi /4} \left(e^{8 \pi }-1\right)}{16 \pi ^{3/2} \sqrt{1+\sqrt2}} \, \Gamma \left(\frac{1}{4}\right)^2.$

Matrix version

Let $A$ be a complex square matrix and positive-definite matrix. Then a $q$ -gamma matrix function can be defined by $q$ -integral:{{cite journal |last1=Salem |first1=Ahmed |date=June 2012 |title=On a q-gamma and a q-beta matrix functions |journal=Linear and Multilinear Algebra |volume=60 |issue=6 |pages=683–696 |doi=10.1080/03081087.2011.627562 | s2cid=123011613 }}

$\Gamma_q(A):=\int_0^{\frac{1}{1-q}}t^{A-I}E_q(-qt)\mathrm{d}_q t$

where $E_q$ is the q-exponential function.

Other ''q''-gamma functions

For other $q$ -gamma functions, see Yamasaki 2006.{{cite journal |last1=Yamasaki |first1=Yoshinori |title=On q-Analogues of the Barnes Multiple Zeta Functions |journal=Tokyo Journal of Mathematics |date=December 2006 |volume=29 |issue=2 |pages=413–427 |arxiv=math/0412067 |doi=10.3836/tjm/1170348176 |mr=2284981 |zbl=1192.11060|s2cid=14082358 }}

Numerical computation

An iterative algorithm to compute the q-gamma function was proposed by Gabutti and Allasia.{{cite journal |last1=Gabutti |first1=Bruno |last2=Allasia |first2=Giampietro |title=Evaluation of q-gamma function and q-analogues by iterative algorithms |journal=Numerical Algorithms |date=17 September 2008 |volume=49 |issue=1–4 |pages=159–168 |doi=10.1007/s11075-008-9196-5|bibcode=2008NuAlg..49..159G |s2cid=6314057 }}

References

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{{citation | last1=El Bachraoui | first1=Mohamed | title=Short proofs for q-Raabe formula and integrals for Jacobi theta functions | year=2017 | journal=Journal of Number Theory | volume=173 | issue=2 | pages=614–620 | doi = 10.1016/j.jnt.2016.09.028 | doi-access=free }}
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Category:Gamma and related functions

Category:Q-analogs