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 Jackson (1905). It is given by

\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 $|q|<1$ , and

\Gamma _{q}(x)={\frac {(q^{-1};q^{-1})_{\infty }}{(q^{-x};q^{-1})_{\infty }}}(q-1)^{1-x}q^{\binom {x}{2}}

if $|q|>1$ . Here ${\displaystyle (\cdot$ 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 (Askey (1978)).
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

\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)).

Transformation Properties

The $q$ -gamma function satisfies the q-analog of the Gauss multiplication formula (Gasper & Rahman (2004)):

\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}^{\infty }{\frac {t^{-z}\mathrm {d} t}{(-t(1-q);q)_{\infty }}}.

Stirling Formula

Moak obtained the following q-analogue of the Stirling formula (see Moak (1984)):

\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 ,

{\hat {q}}=\left\{{\begin{aligned}q\quad \mathrm {if} \ &0<q\leq 1\\1/q\quad \mathrm {if} \ &q\geq 1\end{aligned}}\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=-\infty }^{\infty }\left(r^{m(6m+1)}-r^{(3m+1)(2m+1)}\right),

where $r=\exp(4\pi ^{2}/\log q)$ , $H$ denotes the Heaviside step function, $B_{k}$ stands for the Bernoulli number, $\mathrm {Li} _{2}(z)$ is the dilogarithm, and $p_{k}$ is a polynomial of degree $k$ satisfying

p_{k}(z)=z(1-z)p_{k-1}(z)^{\prime }(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})_{\infty }\quad (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)_{\infty }\quad (0<q<1).

Special values

The following special values are known.[1]

\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}\left(e^{2\pi }-1\right){\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}\left(e^{4\pi }-1\right)}{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}\left(e^{8\pi }-1\right)}{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:[2]

\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.[3]

Numerical Computation

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

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, 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, 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
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

Notes

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

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[1] Mező, István (2011), "Several special values of Jacobi theta functions", arXiv:1106.1042 [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. doi:10.1080/03081087.2011.627562.

[3] Yamasaki, Yoshinori (December 2006). "On q-Analogues of the Barnes Multiple Zeta Functions". Tokyo Journal of Mathematics. 29 (2): 413–427. arXiv:math/0412067. doi:10.3836/tjm/1170348176. MR 2284981. Zbl 1192.11060.

[4] Gabutti, Bruno; Allasia, Giampietro (17 September 2008). "Evaluation of q-gamma function and q-analogues by iterative algorithms". Numerical Algorithms. 49 (1–4): 159–168. Bibcode:2008NuAlg..49..159G. doi:10.1007/s11075-008-9196-5.