- Source: Q-gamma function
In q-analog theory, the
q
{\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
Γ
q
(
x
)
=
(
1
−
q
)
1
−
x
∏
n
=
0
∞
1
−
q
n
+
1
1
−
q
n
+
x
=
(
1
−
q
)
1
−
x
(
q
;
q
)
∞
(
q
x
;
q
)
∞
{\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
|
q
|
<
1
{\displaystyle |q|<1}
, and
Γ
q
(
x
)
=
(
q
−
1
;
q
−
1
)
∞
(
q
−
x
;
q
−
1
)
∞
(
q
−
1
)
1
−
x
q
(
x
2
)
{\displaystyle \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
{\displaystyle |q|>1}
. Here
(
⋅
;
⋅
)
∞
{\displaystyle (\cdot ;\cdot )_{\infty }}
is the infinite q-Pochhammer symbol. The
q
{\displaystyle q}
-gamma function satisfies the functional equation
Γ
q
(
x
+
1
)
=
1
−
q
x
1
−
q
Γ
q
(
x
)
=
[
x
]
q
Γ
q
(
x
)
{\displaystyle \Gamma _{q}(x+1)={\frac {1-q^{x}}{1-q}}\Gamma _{q}(x)=[x]_{q}\Gamma _{q}(x)}
In addition, the
q
{\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,
Γ
q
(
n
)
=
[
n
−
1
]
q
!
{\displaystyle \Gamma _{q}(n)=[n-1]_{q}!}
where
[
⋅
]
q
{\displaystyle [\cdot ]_{q}}
is the q-factorial function. Thus the
q
{\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
lim
q
→
1
±
Γ
q
(
x
)
=
Γ
(
x
)
.
{\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
q
{\displaystyle q}
-gamma function satisfies the q-analog of the Gauss multiplication formula (Gasper & Rahman (2004)):
Γ
q
(
n
x
)
Γ
r
(
1
/
n
)
Γ
r
(
2
/
n
)
⋯
Γ
r
(
(
n
−
1
)
/
n
)
=
(
1
−
q
n
1
−
q
)
n
x
−
1
Γ
r
(
x
)
Γ
r
(
x
+
1
/
n
)
⋯
Γ
r
(
x
+
(
n
−
1
)
/
n
)
,
r
=
q
n
.
{\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
q
{\displaystyle q}
-gamma function has the following integral representation (Ismail (1981)):
1
Γ
q
(
z
)
=
sin
(
π
z
)
π
∫
0
∞
t
−
z
d
t
(
−
t
(
1
−
q
)
;
q
)
∞
.
{\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)):
log
Γ
q
(
x
)
∼
(
x
−
1
/
2
)
log
[
x
]
q
+
L
i
2
(
1
−
q
x
)
log
q
+
C
q
^
+
1
2
H
(
q
−
1
)
log
q
+
∑
k
=
1
∞
B
2
k
(
2
k
)
!
(
log
q
^
q
^
x
−
1
)
2
k
−
1
q
^
x
p
2
k
−
3
(
q
^
x
)
,
x
→
∞
,
{\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 ,}
q
^
=
{
q
i
f
0
<
q
≤
1
1
/
q
i
f
q
≥
1
}
,
{\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\},}
C
q
=
1
2
log
(
2
π
)
+
1
2
log
(
q
−
1
log
q
)
−
1
24
log
q
+
log
∑
m
=
−
∞
∞
(
r
m
(
6
m
+
1
)
−
r
(
3
m
+
1
)
(
2
m
+
1
)
)
,
{\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
r
=
exp
(
4
π
2
/
log
q
)
{\displaystyle r=\exp(4\pi ^{2}/\log q)}
,
H
{\displaystyle H}
denotes the Heaviside step function,
B
k
{\displaystyle B_{k}}
stands for the Bernoulli number,
L
i
2
(
z
)
{\displaystyle \mathrm {Li} _{2}(z)}
is the dilogarithm, and
p
k
{\displaystyle p_{k}}
is a polynomial of degree
k
{\displaystyle k}
satisfying
p
k
(
z
)
=
z
(
1
−
z
)
p
k
−
1
′
(
z
)
+
(
k
z
+
1
)
p
k
−
1
(
z
)
,
p
0
=
p
−
1
=
1
,
k
=
1
,
2
,
⋯
.
{\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
|
q
|
>
1
{\displaystyle |q|>1}
. With this restriction
∫
0
1
log
Γ
q
(
x
)
d
x
=
ζ
(
2
)
log
q
+
log
q
−
1
q
6
+
log
(
q
−
1
;
q
−
1
)
∞
(
q
>
1
)
.
{\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
0
<
q
<
1
{\displaystyle 0<q<1}
and proved that
∫
0
1
log
Γ
q
(
x
)
d
x
=
1
2
log
(
1
−
q
)
−
ζ
(
2
)
log
q
+
log
(
q
;
q
)
∞
(
0
<
q
<
1
)
.
{\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.
Γ
e
−
π
(
1
2
)
=
e
−
7
π
/
16
e
π
−
1
1
+
2
4
2
15
/
16
π
3
/
4
Γ
(
1
4
)
,
{\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),}
Γ
e
−
2
π
(
1
2
)
=
e
−
7
π
/
8
e
2
π
−
1
2
9
/
8
π
3
/
4
Γ
(
1
4
)
,
{\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),}
Γ
e
−
4
π
(
1
2
)
=
e
−
7
π
/
4
e
4
π
−
1
2
7
/
4
π
3
/
4
Γ
(
1
4
)
,
{\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),}
Γ
e
−
8
π
(
1
2
)
=
e
−
7
π
/
2
e
8
π
−
1
2
9
/
4
π
3
/
4
1
+
2
Γ
(
1
4
)
.
{\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
Γ
(
1
2
)
=
π
{\displaystyle \Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }}}
.
Moreover, the following analogues of the familiar identity
Γ
(
1
4
)
Γ
(
3
4
)
=
2
π
{\displaystyle \Gamma \left({\frac {1}{4}}\right)\Gamma \left({\frac {3}{4}}\right)={\sqrt {2}}\pi }
hold true:
Γ
e
−
2
π
(
1
4
)
Γ
e
−
2
π
(
3
4
)
=
e
−
29
π
/
16
(
e
2
π
−
1
)
1
+
2
4
2
33
/
16
π
3
/
2
Γ
(
1
4
)
2
,
{\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},}
Γ
e
−
4
π
(
1
4
)
Γ
e
−
4
π
(
3
4
)
=
e
−
29
π
/
8
(
e
4
π
−
1
)
2
23
/
8
π
3
/
2
Γ
(
1
4
)
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},}
Γ
e
−
8
π
(
1
4
)
Γ
e
−
8
π
(
3
4
)
=
e
−
29
π
/
4
(
e
8
π
−
1
)
16
π
3
/
2
1
+
2
Γ
(
1
4
)
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
A
{\displaystyle A}
be a complex square matrix and Positive-definite matrix. Then a q-gamma matrix function can be defined by q-integral:
Γ
q
(
A
)
:=
∫
0
1
1
−
q
t
A
−
I
E
q
(
−
q
t
)
d
q
t
{\displaystyle \Gamma _{q}(A):=\int _{0}^{\frac {1}{1-q}}t^{A-I}E_{q}(-qt)\mathrm {d} _{q}t}
where
E
q
{\displaystyle 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 Γq(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
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