Q-gamma function

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 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(q1). 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. \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.

References

References

1. Mező, István. (2011). "Several special values of Jacobi theta functions".
2. (June 2012). "On a ''q''-gamma and a ''q''-beta matrix functions". Linear and Multilinear Algebra.
3. (December 2006). "On ''q''-Analogues of the Barnes Multiple Zeta Functions". Tokyo Journal of Mathematics.
4. (17 September 2008). "Evaluation of q-gamma function and q-analogues by iterative algorithms". Numerical Algorithms.