Q-gamma function explained

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$$\backslash Gamma\_q(x)\; =\; (1-q)^\backslash prod\_^\backslash infty\; \backslash frac=(1-q)^\backslash ,\backslash frac$$when |q|< 1 , and$$\backslash Gamma\_q(x)=\backslash frac(q-1)^q^$$if |q|> 1 . Here ( ⋅ ; ⋅ ) _{infty} is the infinite q-Pochhammer symbol. The q -gamma function satisfies the functional equation$$\backslash Gamma\_q(x+1)\; =\; \backslash frac\backslash Gamma\_q(x)=[x]\_q\backslash Gamma\_q(x)$$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,$$\backslash Gamma\_q(n)=[n-1]\_q!$$where [ ⋅ ] _{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$$\backslash lim\_\; \backslash Gamma\_q(x)\; =\; \backslash Gamma(x).$$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 :$$\backslash Gamma\_q(nx)\backslash Gamma\_r(1/n)\backslash Gamma\_r(2/n)\backslash cdots\backslash Gamma\_r((n-1)/n)=\backslash left(\backslash frac\backslash right)^\backslash Gamma\_r(x)\backslash Gamma\_r(x+1/n)\backslash cdots\backslash Gamma\_r(x+(n-1)/n),\; \backslash \; r=q^n.$$

Integral representation

The

q -gamma function has the following integral representation :$$\backslash frac=\backslash frac\backslash int\_0^\backslash infty\backslash frac.$$

Stirling formula

Moak obtained the following q-analogue of the Stirling formula (see):$$\backslash log\backslash Gamma\_q(x)\backslash sim(x-1/2)\backslash log[x]\_q+\backslash frac+C\_+\backslash fracH(q-1)\backslash log\; q+\backslash sum\_^\backslash infty\backslash frac\backslash left(\backslash frac\backslash right)^\backslash hat^x\; p\_(\backslash hat^x),\; \backslash \; x\backslash to\backslash infty,$$$$\backslash hat=\; \backslash left\backslash ,$$$$C\_q\; =\; \backslash frac\; \backslash log(2\backslash pi)+\backslash frac\backslash log\backslash left(\backslash frac\backslash right)-\backslash frac\backslash log\; q+\backslash log\backslash sum\_^\backslash infty\; \backslash left(r^\; -\; r^\backslash right),$$where

r = \exp( 4 \pi ^{2/log} q) , H denotes the Heaviside step function, B _{k} stands for the Bernoulli number, Li _{2(z)} is the dilogarithm, and p _{k} is a polynomial of degree k satisfying$$p\_k(z)=z(1-z)p\text{'}\_(z)+(kz+1)p\_(z),\; p\_0=p\_=1,\; k=1,2,\backslash cdots.$$

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$$\backslash int\_0^1\backslash log\backslash Gamma\_q(x)dx=\backslash frac+\backslash log\backslash sqrt+\backslash log(q^;q^)\_\backslash infty\; \backslash quad(q>1).$$El Bachraoui considered the case 0 <q< 1 and proved that

Special values

The following special values are known.^[1] $$\backslash Gamma\_\backslash left(\backslash frac12\backslash right)=\backslash frac\; \backslash ,\; \backslash Gamma\; \backslash left(\backslash frac\backslash right),$$$$\backslash Gamma\_\backslash left(\backslash frac12\backslash right)=\backslash frac\; \backslash ,\; \backslash Gamma\; \backslash left(\backslash frac\backslash right),$$$$\backslash Gamma\_\backslash left(\backslash frac12\backslash right)=\backslash frac\; \backslash ,\; \backslash Gamma\; \backslash left(\backslash frac\backslash right),$$$$\backslash Gamma\_\backslash left(\backslash frac12\backslash right)=\backslash frac\; \backslash ,\; \backslash Gamma\; \backslash left(\backslash frac\backslash right).$$These are the analogues of the classical formula

\Gamma\left( (12 \right) = \sqrt\pi)/() .

Moreover, the following analogues of the familiar identity

\Gamma\left( (14 \right)\Gamma\left( (34 \right) = \sqrt 2 \pi)/())/() hold true:$$\backslash Gamma\_\backslash left(\backslash frac14\backslash right)\backslash Gamma\_\backslash left(\backslash frac34\backslash right)=\backslash frac\; \backslash ,\; \backslash Gamma\; \backslash left(\backslash frac\backslash right)^2,$$$$\backslash Gamma\_\backslash left(\backslash frac14\backslash right)\backslash Gamma\_\backslash left(\backslash frac34\backslash right)=\backslash frac\; \backslash ,\; \backslash Gamma\; \backslash left(\backslash frac\backslash right)^2,$$$$\backslash Gamma\_\backslash left(\backslash frac14\backslash right)\backslash Gamma\_\backslash left(\backslash frac34\backslash right)=\backslash frac\; \backslash ,\; \backslash Gamma\; \backslash left(\backslash frac\backslash right)^2.$$

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:^[2] $$\backslash Gamma\_q(A):=\backslash int\_0^t^E\_q(-qt)\backslash mathrm\_q\; t$$where E _{q} is the q-exponential function.

Other q-gamma functions

For other q-gamma functions, see Yamasaki 2006.^[3]

Numerical computation

An iterative algorithm to compute the q-gamma function was proposed by Gabutti and Allasia.^[4]

Further reading

• • Zhang . Ruiming . 2010 . On asymptotics of Γ_q(z) as q approaching 1 . 1011.0720 . math.CA . cs2.
• Book: Ismail . Mourad E. H. . Muldoon . Martin E. . Inequalities and monotonicity properties for gamma and q-gamma functions . Zahar . R. V. M. . 1994 . Approximation and computation a festschrift in honor of Walter Gautschi: Proceedings of the Purdue conference, December 2-5, 1993 . 119 . 309–323 . Birkhäuser Verlag . Boston . 1301.1749 . 10.1007/978-1-4684-7415-2_19 . 978-1-4684-7415-2 . 118563435 . cs2.

Notes and References

1. Mező . István . 2011 . Several special values of Jacobi theta functions . 1106.1042 . cs2 . math.NT.
2. Salem . Ahmed . June 2012 . On a q-gamma and a q-beta matrix functions . Linear and Multilinear Algebra . 60 . 6 . 683–696 . 10.1080/03081087.2011.627562 . 123011613 .
3. Yamasaki . Yoshinori . On q-Analogues of the Barnes Multiple Zeta Functions . Tokyo Journal of Mathematics . December 2006 . 29 . 2 . 413–427 . math/0412067 . 10.3836/tjm/1170348176 . 2284981 . 1192.11060. 14082358 .
4. Gabutti . Bruno . Allasia . Giampietro . Evaluation of q-gamma function and q-analogues by iterative algorithms . Numerical Algorithms . 17 September 2008 . 49 . 1–4 . 159–168 . 10.1007/s11075-008-9196-5. 2008NuAlg..49..159G . 6314057 .