q-gamma function

Short description: Function in q-analog theory

In q-analog theory, the [math]\displaystyle{ q }[/math]-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 (Jackson 1905). It is given by [math]\displaystyle{ \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} }[/math] when [math]\displaystyle{ |q|\lt 1 }[/math], and [math]\displaystyle{ \Gamma_q(x)=\frac{(q^{-1};q^{-1})_\infty}{(q^{-x};q^{-1})_\infty}(q-1)^{1-x}q^{\binom{x}{2}} }[/math] if [math]\displaystyle{ |q|\gt 1 }[/math]. Here [math]\displaystyle{ (\cdot;\cdot)_\infty }[/math] is the infinite q-Pochhammer symbol. The [math]\displaystyle{ q }[/math]-gamma function satisfies the functional equation [math]\displaystyle{ \Gamma_q(x+1) = \frac{1-q^{x}}{1-q}\Gamma_q(x)=[x]_q\Gamma_q(x) }[/math] In addition, the [math]\displaystyle{ q }[/math]-gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey ((Askey 1978)).
For non-negative integers n, [math]\displaystyle{ \Gamma_q(n)=[n-1]_q! }[/math] where [math]\displaystyle{ [\cdot]_q }[/math] is the q-factorial function. Thus the [math]\displaystyle{ q }[/math]-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 [math]\displaystyle{ \lim_{q \to 1\pm} \Gamma_q(x) = \Gamma(x). }[/math] There is a simple proof of this limit by Gosper. See the appendix of (Andrews (1986)).

Transformation properties

The [math]\displaystyle{ q }[/math]-gamma function satisfies the q-analog of the Gauss multiplication formula ((Gasper Rahman)): [math]\displaystyle{ \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. }[/math]

Integral representation

The [math]\displaystyle{ q }[/math]-gamma function has the following integral representation (Ismail (1981)): [math]\displaystyle{ \frac{1}{\Gamma_q(z)}=\frac{\sin(\pi z)}{\pi}\int_0^\infty\frac{t^{-z}\mathrm{d}t}{(-t(1-q);q)_{\infty}}. }[/math]

Stirling formula

Moak obtained the following q-analogue of the Stirling formula (see (Moak 1984)): [math]\displaystyle{ \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, }[/math] [math]\displaystyle{ \hat{q}= \left\{\begin{aligned} q \quad \mathrm{if} \ &0\lt q\leq1 \\ 1/q \quad \mathrm{if} \ &q\geq1 \end{aligned}\right\}, }[/math] [math]\displaystyle{ 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), }[/math] where [math]\displaystyle{ r=\exp(4\pi^2/\log q) }[/math], [math]\displaystyle{ H }[/math] denotes the Heaviside step function, [math]\displaystyle{ B_k }[/math] stands for the Bernoulli number, [math]\displaystyle{ \mathrm{Li}_2(z) }[/math] is the dilogarithm, and [math]\displaystyle{ p_k }[/math] is a polynomial of degree [math]\displaystyle{ k }[/math] satisfying [math]\displaystyle{ 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. }[/math]

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 [math]\displaystyle{ |q|\gt 1 }[/math]. With this restriction [math]\displaystyle{ \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\gt 1). }[/math] El Bachraoui considered the case [math]\displaystyle{ 0\lt q\lt 1 }[/math] and proved that [math]\displaystyle{ \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\lt q\lt 1). }[/math]

Special values

The following special values are known. [math]\displaystyle{ \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), }[/math] [math]\displaystyle{ \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), }[/math] [math]\displaystyle{ \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), }[/math] [math]\displaystyle{ \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). }[/math] These are the analogues of the classical formula [math]\displaystyle{ \Gamma\left(\frac12\right)=\sqrt\pi }[/math].

Moreover, the following analogues of the familiar identity [math]\displaystyle{ \Gamma\left(\frac14\right)\Gamma\left(\frac34\right)=\sqrt2\pi }[/math] hold true: [math]\displaystyle{ \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, }[/math] [math]\displaystyle{ \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, }[/math] [math]\displaystyle{ \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. }[/math]

Matrix Version

Let [math]\displaystyle{ A }[/math] be a complex square matrix and Positive-definite matrix. Then a q-gamma matrix function can be defined by q-integral: [math]\displaystyle{ \Gamma_q(A):=\int_0^{\frac{1}{1-q}}t^{A-I}E_q(-qt)\mathrm{d}_q t }[/math] where [math]\displaystyle{ E_q }[/math] 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

• Zhang, Ruiming (2007), "On asymptotics of q-gamma functions", Journal of Mathematical Analysis and Applications 339 (2): 1313–1321, doi:10.1016/j.jmaa.2007.08.006, Bibcode: 2008JMAA..339.1313Z 
• Zhang, Ruiming (2010), "On asymptotics of Γ_q(z) as q approaching 1", arXiv:1011.0720 [math.CA]
• Ismail, Mourad E. H.; Muldoon, Martin E. (1994). "Inequalities and monotonicity properties for gamma and q-gamma functions". in Zahar, R. V. M.. Approximation and computation a festschrift in honor of Walter Gautschi: Proceedings of the Purdue conference, December 2-5, 1993. 119. Boston: Birkhäuser Verlag. pp. 309–323. doi:10.1007/978-1-4684-7415-2_19. ISBN 978-1-4684-7415-2. 

References

• Jackson, F. H. (1905), "The Basic Gamma-Function and the Elliptic Functions", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character (The Royal Society) 76 (508): 127–144, doi:10.1098/rspa.1905.0011, ISSN 0950-1207, Bibcode: 1905RSPSA..76..127J 
• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8 
• Ismail, Mourad (1981), "The Basic Bessel Functions and Polynomials", SIAM Journal on Mathematical Analysis 12 (3): 454–468, doi:10.1137/0512038 
• Moak, Daniel S. (1984), "The Q-analogue of Stirling's formula", Rocky Mountain J. Math. 14 (2): 403–414, doi:10.1216/RMJ-1984-14-2-403 
• Mező, István (2012), "A q-Raabe formula and an integral of the fourth Jacobi theta function", Journal of Number Theory 133 (2): 692–704, doi:10.1016/j.jnt.2012.08.025 
• El Bachraoui, Mohamed (2017), "Short proofs for q-Raabe formula and integrals for Jacobi theta functions", Journal of Number Theory 173 (2): 614–620, doi:10.1016/j.jnt.2016.09.028 
• Askey, Richard (1978), "The q-gamma and q-beta functions.", Applicable Analysis 8 (2): 125–141, doi:10.1080/00036817808839221 
• Andrews, George E. (1986), q-Series: Their development and application in analysis, number theory, combinatorics, physics, and computer algebra., Regional Conference Series in Mathematics, 66, American Mathematical Society