# 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 q-Pochhammer symbol. The ${\displaystyle q}$-gamma function satisfies the functional equation

${\displaystyle \Gamma _{q}(x+1)={\frac {1-q^{x}}{1-q}}\Gamma _{q}(x)=[x]_{q}\Gamma _{q}(x)}$

For non-negative integers n,

${\displaystyle \Gamma _{q}(n)=[n-1]_{q}!}$

where ${\displaystyle [\cdot ]_{q}}$ is the q-factorial function. Thus the ${\displaystyle q}$-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

${\displaystyle \lim _{q\to 1\pm }\Gamma _{q}(x)=\Gamma (x).}$

## 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 ${\displaystyle |q|>1}$. 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 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

## Special values

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:

${\displaystyle \Gamma _{e^{-2\pi }}\left({\frac {1}{4}}\right)\Gamma _{e^{-2\pi }}\left({\frac {3}{4}}\right)={\frac {e^{-29\pi /16}\left(e^{2\pi }-1\right){\sqrt[{4}]{1+{\sqrt {2}}}}}{2^{33/16}\pi ^{3/2}}}\,\Gamma \left({\frac {1}{4}}\right)^{2},}$
${\displaystyle \Gamma _{e^{-4\pi }}\left({\frac {1}{4}}\right)\Gamma _{e^{-4\pi }}\left({\frac {3}{4}}\right)={\frac {e^{-29\pi /8}\left(e^{4\pi }-1\right)}{2^{23/8}\pi ^{3/2}}}\,\Gamma \left({\frac {1}{4}}\right)^{2},}$
${\displaystyle \Gamma _{e^{-8\pi }}\left({\frac {1}{4}}\right)\Gamma _{e^{-8\pi }}\left({\frac {3}{4}}\right)={\frac {e^{-29\pi /4}\left(e^{8\pi }-1\right)}{16\pi ^{3/2}{\sqrt {1+{\sqrt {2}}}}}}\,\Gamma \left({\frac {1}{4}}\right)^{2}.}$

## 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, ISSN 0950-1207, JSTOR 92601, doi:10.1098/rspa.1905.0011
• 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, MR 2128719
• 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
Notes
1. ^ Mező, István, Several special values of Jacobi theta functions.