q-gamma function

In q-analog theory, the ${\displaystyle q}$-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 Jackson (1905). It is given by $${\displaystyle \Gamma _{q}(x)=(1-q)^{1-x}\prod _{n=0}^{\infty }{\frac {1-q^{n+1}}{1-q^{n+x}}}=(1-q)^{1-x}\,{\frac {(q;q)_{\infty }}{(q^{x};q)_{\infty }}}}$$ when ${\displaystyle |q|<1}$, and $${\displaystyle \Gamma _{q}(x)={\frac {(q^{-1};q^{-1})_{\infty }}{(q^{-x};q^{-1})_{\infty }}}(q-1)^{1-x}q^{\binom {x}{2}}}$$ if ${\displaystyle |q|>1}$. Here ${\displaystyle (\cdot ;\cdot )_{\infty }}$ is the infinite ${\displaystyle q}$-Pochhammer symbol. The ${\displaystyle q}$-gamma function satisfies the functional equation $${\displaystyle \Gamma _{q}(x+1)={\frac {1-q^{x}}{1-q}}\Gamma _{q}(x)=[x]_{q}\Gamma _{q}(x)}$$ In addition, the ${\displaystyle q}$-gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey (Askey (1978)).

For non-negative integers ${\displaystyle n}$, $${\displaystyle \Gamma _{q}(n)=[n-1]_{q}!}$$ where ${\displaystyle [\cdot ]_{q}}$ is the ${\displaystyle q}$-factorial function. Thus the ${\displaystyle q}$-gamma function can be considered as an extension of the ${\displaystyle q}$-factorial function to the real numbers.

The relation to the ordinary gamma function is made explicit in the limit $${\displaystyle \lim _{q\to 1\pm }\Gamma _{q}(x)=\Gamma (x).}$$ There is a simple proof of this limit by Gosper. See the appendix of (Andrews (1986)).

The ${\displaystyle q}$-gamma function satisfies the q-analog of the Gauss multiplication formula (Gasper & Rahman (2004)): $${\displaystyle \Gamma _{q}(nx)\Gamma _{r}(1/n)\Gamma _{r}(2/n)\cdots \Gamma _{r}((n-1)/n)=\left({\frac {1-q^{n}}{1-q}}\right)^{nx-1}\Gamma _{r}(x)\Gamma _{r}(x+1/n)\cdots \Gamma _{r}(x+(n-1)/n),\ r=q^{n}.}$$

The ${\displaystyle q}$-gamma function has the following integral representation (Ismail (1981)): $${\displaystyle {\frac {1}{\Gamma _{q}(z)}}={\frac {\sin(\pi z)}{\pi }}\int _{0}^{\infty }{\frac {t^{-z}\mathrm {d} t}{(-t(1-q);q)_{\infty }}}.}$$

Moak obtained the following q-analogue of the Stirling formula (see Moak (1984)): $${\displaystyle \log \Gamma _{q}(x)\sim (x-1/2)\log[x]_{q}+{\frac {\mathrm {Li} _{2}(1-q^{x})}{\log q}}+C_{\hat {q}}+{\frac {1}{2}}H(q-1)\log q+\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}\left({\frac {\log {\hat {q}}}{{\hat {q}}^{x}-1}}\right)^{2k-1}{\hat {q}}^{x}p_{2k-3}({\hat {q}}^{x}),\ x\to \infty ,}$$ $${\displaystyle {\hat {q}}=\left\{{\begin{aligned}q\quad \mathrm {if} \ &0<q\leq 1\\1/q\quad \mathrm {if} \ &q\geq 1\end{aligned}}\right\},}$$ $${\displaystyle C_{q}={\frac {1}{2}}\log(2\pi )+{\frac {1}{2}}\log \left({\frac {q-1}{\log q}}\right)-{\frac {1}{24}}\log q+\log \sum _{m=-\infty }^{\infty }\left(r^{m(6m+1)}-r^{(3m+1)(2m+1)}\right),}$$ where ${\displaystyle r=\exp(4\pi ^{2}/\log q)}$, ${\displaystyle H}$ denotes the Heaviside step function, ${\displaystyle B_{k}}$ stands for the Bernoulli number, ${\displaystyle \mathrm {Li} _{2}(z)}$ is the dilogarithm, and ${\displaystyle p_{k}}$ is a polynomial of degree ${\displaystyle k}$ satisfying $${\displaystyle p_{k}(z)=z(1-z)p'_{k-1}(z)+(kz+1)p_{k-1}(z),p_{0}=p_{-1}=1,k=1,2,\cdots .}$$

Due to I. Mező, the q-analogue of the Raabe formula exists, at least if we use the ${\displaystyle q}$-gamma function when ${\displaystyle |q|>1}$. With this restriction, $${\displaystyle \int _{0}^{1}\log \Gamma _{q}(x)dx={\frac {\zeta (2)}{\log q}}+\log {\sqrt {\frac {q-1}{\sqrt[{6}]{q}}}}+\log(q^{-1};q^{-1})_{\infty }\quad (q>1).}$$ El Bachraoui considered the case ${\displaystyle 0<q<1}$ and proved that $${\displaystyle \int _{0}^{1}\log \Gamma _{q}(x)dx={\frac {1}{2}}\log(1-q)-{\frac {\zeta (2)}{\log q}}+\log(q;q)_{\infty }\quad (0<q<1).}$$

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

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

Let ${\displaystyle A}$ be a complex square matrix and positive-definite matrix. Then a ${\displaystyle q}$-gamma matrix function can be defined by ${\displaystyle q}$-integral:^[2] $${\displaystyle \Gamma _{q}(A):=\int _{0}^{\frac {1}{1-q}}t^{A-I}E_{q}(-qt)\mathrm {d} _{q}t}$$ where ${\displaystyle E_{q}}$ is the q-exponential function.

For other ${\displaystyle q}$-gamma functions, see Yamasaki 2006.^[3]

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

• 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

• 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