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. 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. 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.

Transformation properties
The $$q$$-gamma function satisfies the q-analog of the Gauss multiplication formula : $$\Gamma_q(nx)\Gamma_r(1/n)\Gamma_r(2/n)\cdots\Gamma_r((n-1)/n)=\left(\frac{1-q^n}{1-q}\right)^{nx-1}\Gamma_r(x)\Gamma_r(x+1/n)\cdots\Gamma_r(x+(n-1)/n), \ r=q^n.$$

Integral representation
The $$q$$-gamma function has the following integral representation : $$\frac{1}{\Gamma_q(z)}=\frac{\sin(\pi z)}{\pi}\int_0^\infty\frac{t^{-z}\mathrm{d}t}{(-t(1-q);q)_{\infty}}.$$

Stirling formula
Moak obtained the following q-analogue of the Stirling formula (see ): $$\log\Gamma_q(x)\sim(x-1/2)\log[x]_q+\frac{\mathrm{Li}_2(1-q^x)}{\log q}+C_{\hat{q}}+\frac{1}{2}H(q-1)\log q+\sum_{k=1}^\infty \frac{B_{2k}}{(2k)!}\left(\frac{\log \hat{q}}{\hat{q}^x-1}\right)^{2k-1}\hat{q}^x p_{2k-3}(\hat{q}^x), \ x\to\infty,$$ $$\hat{q}= \left\{\begin{aligned} q \quad \mathrm{if} \ &0<q\leq1 \\ 1/q \quad \mathrm{if} \ &q\geq1 \end{aligned}\right\},$$ $$C_q = \frac{1}{2} \log(2\pi)+\frac{1}{2}\log\left(\frac{q-1}{\log q}\right)-\frac{1}{24}\log q+\log\sum_{m=-\infty}^\infty \left(r^{m(6m+1)} - r^{(3m+1)(2m+1)}\right),$$ 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 $$0<q<1$$ and proved that $$ \int_0^1\log\Gamma_q(x)dx=\frac{1}{2}\log (1-q) - \frac{\zeta(2)}{\log q}+\log(q;q)_\infty \quad(0<q<1). $$

Special values
The following special values are known. $$\Gamma_{e^{-\pi}}\left(\frac12\right)=\frac{e^{-7 \pi /16} \sqrt{e^\pi-1}\sqrt[4]{1+\sqrt2}}{2^{15/16}\pi^{3/4}} \, \Gamma \left(\frac{1}{4}\right),$$ $$\Gamma_{e^{-2\pi}}\left(\frac12\right)=\frac{e^{-7 \pi /8} \sqrt{e^{2 \pi}-1}}{2^{9/8} \pi^{3/4}} \, \Gamma \left(\frac{1}{4}\right),$$ $$\Gamma_{e^{-4\pi}}\left(\frac12\right)=\frac{e^{-7 \pi /4} \sqrt{e^{4 \pi}-1}}{2^{7/4} \pi^{3/4}} \, \Gamma \left(\frac{1}{4}\right),$$ $$\Gamma_{e^{-8\pi}}\left(\frac12\right)=\frac{e^{-7 \pi /2} \sqrt{e^{8 \pi}-1}}{2^{9/4} \pi^{3/4} \sqrt{1+\sqrt2}} \, \Gamma \left(\frac{1}{4}\right).$$ 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: $$\Gamma_q(A):=\int_0^{\frac{1}{1-q}}t^{A-I}E_q(-qt)\mathrm{d}_q t $$ where $$E_q$$ is the q-exponential function.

Other q-gamma functions
For other q-gamma functions, see Yamasaki 2006.

Numerical computation
An iterative algorithm to compute the q-gamma function was proposed by Gabutti and Allasia.