In mathematics, the indefinite product operator is the inverse operator of
Q
(
f
(
x
)
)
=
f
(
x
+
1
)
f
(
x
)
. It is like a discrete version of the indefinite product integral. Some authors use term discrete multiplicative integration
Thus
Q
(
∏
x
f
(
x
)
)
=
f
(
x
)
.
More explicitly, if
∏
x
f
(
x
)
=
F
(
x
)
, then
F
(
x
+
1
)
F
(
x
)
=
f
(
x
)
.
If F(x) is a solution of this functional equation for a given f(x), then so is CF(x) for any constant C. Therefore each indefinite product actually represents a family of functions, differing by a multiplicative constant.
If
T
is a period of function
f
(
x
)
then
∏
x
f
(
T
x
)
=
C
f
(
T
x
)
x
−
1
Indefinite product can be expressed in terms of indefinite sum:
∏
x
f
(
x
)
=
exp
(
∑
x
ln
f
(
x
)
)
Some authors use the phrase "indefinite product" in a slightly different but related way to describe a product in which the numerical value of the upper limit is not given. e.g.
∏
k
=
1
n
f
(
k
)
.
∏
x
f
(
x
)
g
(
x
)
=
∏
x
f
(
x
)
∏
x
g
(
x
)
∏
x
f
(
x
)
a
=
(
∏
x
f
(
x
)
)
a
∏
x
a
f
(
x
)
=
a
∑
x
f
(
x
)
This is a list of indefinite products
∏
x
f
(
x
)
. Not all functions have an indefinite product which can be expressed in elementary functions.
∏
x
a
=
C
a
x
∏
x
x
=
C
Γ
(
x
)
∏
x
x
+
1
x
=
C
x
∏
x
x
+
a
x
=
C
Γ
(
x
+
a
)
Γ
(
x
)
∏
x
x
a
=
C
Γ
(
x
)
a
∏
x
a
x
=
C
a
x
Γ
(
x
)
∏
x
a
x
=
C
a
x
2
(
x
−
1
)
∏
x
a
1
x
=
C
a
Γ
′
(
x
)
Γ
(
x
)
∏
x
x
x
=
C
e
ζ
′
(
−
1
,
x
)
−
ζ
′
(
−
1
)
=
C
e
ψ
(
−
2
)
(
z
)
+
z
2
−
z
2
−
z
2
ln
(
2
π
)
=
C
K
(
x
)
(see K-function)
∏
x
Γ
(
x
)
=
C
Γ
(
x
)
x
−
1
K
(
x
)
=
C
Γ
(
x
)
x
−
1
e
z
2
ln
(
2
π
)
−
z
2
−
z
2
−
ψ
(
−
2
)
(
z
)
=
C
G
(
x
)
(see Barnes G-function)
∏
x
sexp
a
(
x
)
=
C
(
sexp
a
(
x
)
)
′
sexp
a
(
x
)
(
ln
a
)
x
(see super-exponential function)
∏
x
x
+
a
=
C
Γ
(
x
+
a
)
∏
x
a
x
+
b
=
C
a
x
Γ
(
x
+
b
a
)
∏
x
a
x
2
+
b
x
=
C
a
x
Γ
(
x
)
Γ
(
x
+
b
a
)
∏
x
x
2
+
1
=
C
Γ
(
x
−
i
)
Γ
(
x
+
i
)
∏
x
x
+
1
x
=
C
Γ
(
x
−
i
)
Γ
(
x
+
i
)
Γ
(
x
)
∏
x
csc
x
sin
(
x
+
1
)
=
C
sin
x
∏
x
sec
x
cos
(
x
+
1
)
=
C
cos
x
∏
x
cot
x
tan
(
x
+
1
)
=
C
tan
x
∏
x
tan
x
cot
(
x
+
1
)
=
C
cot
x
