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

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

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 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

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

There is a simple proof of this limit by Gosper. See the appendix of (Andrews  ( 1986 )).

Transformation properties

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

$${\displaystyle \Gamma _{q}(nx)\Gamma _{r}(1/n)\Gamma _{r}(2/n)\cdots \Gamma _{r}((n-1)/n)=\left({\frac {1-q^{n}}{1-q}}\right)^{nx-1}\Gamma _{r}(x)\Gamma _{r}(x+1/n)\cdots \Gamma _{r}(x+(n-1)/n),\ r=q^{n}.}$$

Integral representation

The ${\displaystyle q}$-gamma function has the following integral representation (Ismail  ( 1981 )):

$${\displaystyle {\frac {1}{\Gamma _{q}(z)}}={\frac {\sin(\pi z)}{\pi }}\int _{0}^{\infty }{\frac {t^{-z}\mathrm {d} t}{(-t(1-q);q)_{\infty }}}.}$$

Stirling formula

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

$${\displaystyle p_{k}(z)=z(1-z)p'_{k-1}(z)+(kz+1)p_{k-1}(z),p_{0}=p_{-1}=1,k=1,2,\cdots .}$$

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 ${\displaystyle 0<q<1}$ and proved that

$${\displaystyle \int _{0}^{1}\log \Gamma _{q}(x)dx={\frac {1}{2}}\log(1-q)-{\frac {\zeta (2)}{\log q}}+\log(q;q)_{\infty }\quad (0<q<1).}$$

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:

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

$${\displaystyle \Gamma _{e^{-4\pi }}\left({\frac {1}{4}}\right)\Gamma _{e^{-4\pi }}\left({\frac {3}{4}}\right)={\frac {e^{-29\pi /8}\left(e^{4\pi }-1\right)}{2^{23/8}\pi ^{3/2}}}\,\Gamma \left({\frac {1}{4}}\right)^{2},}$$

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

Matrix Version

Let ${\displaystyle A}$ be a complex square matrix and Positive-definite matrix. Then a q-gamma matrix function can be defined by 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.

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]

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

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. S2CID   123011613.
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. S2CID   14082358. 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. S2CID   6314057.

• 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, 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
• 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