In mathematics, the Retkes Identities, named after Zoltán Retkes, are one of the most efficient applications of the Retkes inequality, when
f
(
u
)
=
u
α
,
0
≤
u
<
∞
, and
0
≤
α
. In this special setting, one can have for the iterated integrals
F
(
n
−
1
)
(
s
)
=
s
α
+
n
−
1
(
α
+
1
)
(
α
+
2
)
⋯
(
α
+
n
−
1
)
.
The notation is explained at Hermite–Hadamard inequality.
Since
f
is strictly convex if
α
>
1
, strictly concave if
0
<
α
<
1
, linear if
α
=
0
,
1
, the following inequalities and identities hold:
1
<
α
1
(
α
+
1
)
(
α
+
2
)
⋯
(
α
+
n
−
1
)
∑
i
=
1
n
x
i
α
+
n
−
1
Π
k
(
x
1
,
…
,
x
n
)
<
1
n
!
∑
i
=
1
n
x
i
α
α
=
1
∑
i
=
1
n
x
i
n
Π
i
(
x
1
,
…
,
x
n
)
=
∑
i
=
1
n
x
i
0
<
α
<
1
1
(
α
+
1
)
(
α
+
2
)
⋯
(
α
+
n
−
1
)
∑
i
=
1
n
x
i
α
+
n
−
1
Π
k
(
x
1
,
…
,
x
n
)
>
1
n
!
∑
i
=
1
n
x
i
α
α
=
0
∑
i
=
1
n
x
i
n
−
1
Π
i
(
x
1
,
…
,
x
n
)
=
1.
One of the consequences of the case
α
=
1
is the Retkes convergence criterion because of the right side of the equality is precisely the nth partial sum of
∑
i
=
1
∞
x
i
.
Assume henceforth that
x
k
≠
0
k
=
1
,
…
,
n
.
Under this condition substituting
1
x
k
instead of
x
k
in the second and fourth identities one can have two universal algebraic identities. These four identities are the so-called Retkes identities:
∑
i
=
1
n
x
i
n
Π
i
(
x
1
,
…
,
x
n
)
=
∑
i
=
1
n
x
i
∑
i
=
1
n
x
i
n
−
1
Π
i
(
x
1
,
…
,
x
n
)
=
1
∑
i
=
1
n
1
x
i
=
(
−
1
)
n
−
1
∏
i
=
1
n
x
i
∑
i
=
1
n
1
x
i
2
Π
i
(
x
1
,
…
,
x
n
)
∏
i
=
1
n
1
x
i
=
(
−
1
)
n
−
1
∑
i
=
1
n
1
x
i
Π
i
(
x
1
,
…
,
x
n
)
