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
$\Gamma_q(x) = (1-q)^\prod_^\infty \frac=(1-q)^\,\frac$
when $, q, <1$, and
$\Gamma_q(x)=\frac(q-1)^q^$
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\Gamma_q(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)= -1q!
where $$cdotq 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
$\lim_ \Gamma_q(x) = \Gamma(x).$
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 ():
$\Gamma_q(nx)\Gamma_r(1/n)\Gamma_r(2/n)\cdots\Gamma_r((n-1)/n)=\left(\frac\right)^\Gamma_r(x)\Gamma_r(x+1/n)\cdots\Gamma_r(x+(n-1)/n), \ r=q^n.$

Integral representation

The $q$-gamma function has the following integral representation ():
$\frac=\frac\int_0^\infty\frac.$

Stirling formula

Moak obtained the following q-analogue of the Stirling formula (see ):
\log\Gamma_q(x)\sim(x-1/2)\log q+\frac+C_+\fracH(q-1)\log q+\sum_^\infty
\frac\left(\frac\right)^\hat^x p_(\hat^x), \ x\to\infty,
$\hat= \left\,$
$C_q = \frac \log(2\pi)+\frac\log\left(\frac\right)-\frac\log q+\log\sum_^\infty \left(r^ - r^\right),$
where $r=\exp(4\pi^2/\log q)$, $H$ denotes the Heaviside step function, $B_k$ stands for the Bernoulli number, $\mathrm_2(z)$ is the dilogarithm, and $p_k$ is a polynomial of degree $k$ satisfying
$p_k(z)=z(1-z)p'_(z)+(kz+1)p_(z), p_0=p_=1, k=1,2,\cdots.$

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
$\int_0^1\log\Gamma_q(x)dx=\frac+\log\sqrt+\log(q^;q^)_\infty \quad(q>1).$
El Bachraoui considered the case 0 and proved that
\int_0^1\log\Gamma_q(x)dx=\frac\log (1-q) - \frac+\log(q;q)_\infty \quad(0

Special values

The following special values are known.
$\Gamma_\left(\frac12\right)=\frac \, \Gamma \left(\frac\right),$
$\Gamma_\left(\frac12\right)=\frac \, \Gamma \left(\frac\right),$
$\Gamma_\left(\frac12\right)=\frac \, \Gamma \left(\frac\right),$
$\Gamma_\left(\frac12\right)=\frac \, \Gamma \left(\frac\right).$
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:
$\Gamma_\left(\frac14\right)\Gamma_\left(\frac34\right)=\frac \, \Gamma \left(\frac\right)^2,$
$\Gamma_\left(\frac14\right)\Gamma_\left(\frac34\right)=\frac \, \Gamma \left(\frac\right)^2,$
$\Gamma_\left(\frac14\right)\Gamma_\left(\frac34\right)=\frac \, \Gamma \left(\frac\right)^2.$

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:
$\Gamma_q(A):=\int_0^t^E_q(-qt)\mathrm_q t$
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.

Further reading

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References

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*{{Citation , last1=Andrews , first1=George E. , title=q-Series: Their development and application in analysis, number theory, combinatorics, physics, and computer algebra. , year=1986 , publisher=American Mathematical Society , series=Regional Conference Series in Mathematics , volume=66

Gamma and related functions
Q-analogs