q
-Gamma and Beta Functions
-
q
-Factorials
-
q
-Gamma Function
-
q
-Beta Function
q
-Factorials
(
a
;
q
)
n
=
∏
k
=
0
n
-
1
(
1
-
a
q
k
)
n
=
0
,
1
,
2
,
…
n
!
q
=
1
(
1
+
q
)
⋯
(
1
+
q
+
…
+
q
n
-
1
)
=
(
q
;
q
)
n
(
1
-
q
)
-
n
When
|
q
|
<
1
,
(
a
;
q
)
∞
=
∏
k
=
0
∞
(
1
-
a
q
k
)
q-Gamma Function
When
0
<
q
<
1
,
Γ
q
(
z
)
=
(
q
;
q
)
∞
(
1
-
q
)
1
-
z
(
q
z
;
q
)
∞
Γ
q
(
1
)
=
Γ
q
(
2
)
=
1
n
!
q
=
Γ
q
(
n
+
1
)
Γ
q
(
z
+
1
)
=
1
-
q
z
1
-
q
Γ
q
(
z
)
Also,
ln
Γ
q
(
x
)
is convex for
x
>
0
, and the analog of the
Bohr-Mollerup theorem
holds.
If
0
<
q
<
r
<
1
, then
Γ
q
(
x
)
<
Γ
r
(
x
)
when
0
<
x
<
1
or when
x
>
2
, and
Γ
q
(
x
)
>
Γ
r
(
x
)
when
1
<
x
<
2
.
lim
q
→
1
-
Γ
q
(
z
)
=
Γ
(
z
)
For generalized asymptotic expansions of
ln
Γ
q
(
z
)
as
|
z
|
→
∞
see
Olde Daalhuis(1994)
and
Moak(1984)
.
q
-Beta Function
B
q
(
a
,
b
)
=
Γ
q
(
a
)
Γ
q
(
b
)
Γ
q
(
a
+
b
)
B
q
(
a
,
b
)
=
∫
0
1
t
a
-
1
(
t
q
;
q
)
∞
(
t
q
b
;
q
)
∞
ⅆ
q
t
0
<
q
<
1
,
ℜ
a
>
0
,
ℜ
b
>
0
.