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 $${\Gamma }_q(x)=(1-q{)}^{1-x}\prod _{n=0}^{\mathrm{\infty }}\frac{1-q^{n+1}}{1-q^{n+x}}=(1-q{)}^{1-x}\frac{(q;q{)}_{\mathrm{\infty }}}{(q^x;q{)}_{\mathrm{\infty }}}$$ when $|q|<1$, and $${\Gamma }_q(x)=\frac{(q^{-1};q^{-1}{)}_{\mathrm{\infty }}}{(q^{-x};q^{-1}{)}_{\mathrm{\infty }}}(q-1{)}^{1-x}{q}^{(\genfrac{}{}{0ex}{}{x}{2})}$$ if $|q|>1$. Here $(\cdot ;\cdot {)}_{\mathrm{\infty }}$ 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 $$\underset{q\to 1\pm }{lim}{\Gamma }_q(x)=\Gamma (x).$$ There is a simple proof of this limit by Gosper. See the appendix of (Andrews (1986)).

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 (Ismail (1981)): $$\frac{1}{{\Gamma }_q(z)}=\frac{\sin (\pi z)}{\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-z}\mathrm{d}t}{(-t(1-q);q{)}_{\mathrm{\infty }}}.$$

Stirling formula

Moak obtained the following q-analogue of the Stirling formula (see ): $$\log {\Gamma }_q(x)\sim (x-1/2)\log [x{]}_q+\frac{{\mathrm{L}\mathrm{i}}_2(1-q^x)}{\log q}+C_{\hat{q}}+\frac{1}{2}H(q-1)\log q+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{(2k)!}{\left(\frac{\log \hat{q}}{{\hat{q}}^x-1}\right)}^{2k-1}{\hat{q}}^x{p}_{2k-3}({\hat{q}}^x),x\to \mathrm{\infty },$$ $$\hat{q}=\left\{\begin{array}{cc}q\mathrm{i}\mathrm{f}& 0<q\le 1\\ 1/q\mathrm{i}\mathrm{f}& q\ge 1\end{array}\right\},$$ $$C_q=\frac{1}{2}\log (2\pi )+\frac{1}{2}\log \left(\frac{q-1}{\log q}\right)-\frac{1}{24}\log q+\log \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}(r^{m(6m+1)}-r^{(3m+1)(2m+1)}),$$ where $r=\exp (4{\pi }^2/\log q)$, $H$ denotes the Heaviside step function, $B_k$ stands for the Bernoulli number, ${\mathrm{L}\mathrm{i}}_2(z)$ is the dilogarithm, and $p_k$ is a polynomial of degree $k$ satisfying $$p_k(z)=z(1-z)p{\text{'}}_{k-1}(z)+(kz+1)p_{k-1}(z),p_0=p_{-1}=1,k=1,2,\cdots .$$

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, $${\int }_0^1\log {\Gamma }_q(x)dx=\frac{\zeta (2)}{\log q}+\log \sqrt{\frac{q-1}{\sqrt[6]{q}}}+\log (q^{-1};q^{-1}{)}_{\mathrm{\infty }}(q>1).$$ El Bachraoui considered the case $0<q<1$ and proved that $${\int }_0^1\log {\Gamma }_q(x)dx=\frac{1}{2}\log (1-q)-\frac{\zeta (2)}{\log q}+\log (q;q{)}_{\mathrm{\infty }}(0<q<1).$$

Special values

The following special values are known. $${\Gamma }_{e^{-\pi }}\left(\frac{1}{2}\right)=\frac{e^{-7\pi /16}\sqrt{e^{\pi }-1}\sqrt[4]{1+\sqrt{2}}}{2^{15/16}{\pi }^{3/4}}\Gamma \left(\frac{1}{4}\right),$$ $${\Gamma }_{e^{-2\pi }}\left(\frac{1}{2}\right)=\frac{e^{-7\pi /8}\sqrt{e^{2\pi }-1}}{2^{9/8}{\pi }^{3/4}}\Gamma \left(\frac{1}{4}\right),$$ $${\Gamma }_{e^{-4\pi }}\left(\frac{1}{2}\right)=\frac{e^{-7\pi /4}\sqrt{e^{4\pi }-1}}{2^{7/4}{\pi }^{3/4}}\Gamma \left(\frac{1}{4}\right),$$ $${\Gamma }_{e^{-8\pi }}\left(\frac{1}{2}\right)=\frac{e^{-7\pi /2}\sqrt{e^{8\pi }-1}}{2^{9/4}{\pi }^{3/4}\sqrt{1+\sqrt{2}}}\Gamma \left(\frac{1}{4}\right).$$ These are the analogues of the classical formula $\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }$.

Moreover, the following analogues of the familiar identity $\Gamma \left(\frac{1}{4}\right)\Gamma \left(\frac{3}{4}\right)=\sqrt{2}\pi$ hold true: $${\Gamma }_{e^{-2\pi }}\left(\frac{1}{4}\right){\Gamma }_{e^{-2\pi }}\left(\frac{3}{4}\right)=\frac{e^{-29\pi /16}(e^{2\pi }-1)\sqrt[4]{1+\sqrt{2}}}{2^{33/16}{\pi }^{3/2}}\Gamma {\left(\frac{1}{4}\right)}^2,$$ $${\Gamma }_{e^{-4\pi }}\left(\frac{1}{4}\right){\Gamma }_{e^{-4\pi }}\left(\frac{3}{4}\right)=\frac{e^{-29\pi /8}(e^{4\pi }-1)}{2^{23/8}{\pi }^{3/2}}\Gamma {\left(\frac{1}{4}\right)}^2,$$ $${\Gamma }_{e^{-8\pi }}\left(\frac{1}{4}\right){\Gamma }_{e^{-8\pi }}\left(\frac{3}{4}\right)=\frac{e^{-29\pi /4}(e^{8\pi }-1)}{16{\pi }^{3/2}\sqrt{1+\sqrt{2}}}\Gamma {\left(\frac{1}{4}\right)}^2.$$

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.

Further reading

• Zhang, Ruiming (2007), "On asymptotics of q-gamma functions", Journal of Mathematical Analysis and Applications, 339 (2): 1313–1321, arXiv:0705.2802, Bibcode:2008JMAA..339.1313Z, doi:10.1016/j.jmaa.2007.08.006, S2CID 115163047
• 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. (ed.), Approximation and computation a festschrift in honor of Walter Gautschi: Proceedings of the Purdue conference, December 2-5, 1993, vol. 119, Boston: Birkhäuser Verlag, pp. 309–323, arXiv:1301.1749, doi:10.1007/978-1-4684-7415-2_19, ISBN 978-1-4684-7415-2, S2CID 118563435

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, 76 (508), The Royal Society: 127–144, Bibcode:1905RSPSA..76..127J, doi:10.1098/rspa.1905.0011, ISSN 0950-1207, JSTOR 92601
• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719
• 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, hdl:2437/166217
• 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, vol. 66, American Mathematical Society