Polygamma Functions
The functions
ψ
(
n
)
(
z
)
,
n
=
1
,
2
,
…
, are called the polygamma functions. In particular,
ψ
′
(
z
)
is the trigamma function;
ψ
′
′
,
ψ
(
3
)
,
ψ
(
4
)
are the tetra-, penta-, and
hexagamma functions respectively. Most properties of these
functions follow straightforwardly by differentiation of properties
of the psi function. This includes asymptotic expansions.
In the second and third equations,
n
=
1
,
2
,
3
,
…
; for
ζ
(
n
+
1
)
ψ
′
(
z
)
=
∑
k
=
0
∞
1
(
k
+
z
)
2
z
≠
0
,
-
1
,
-
2
,
…
ψ
(
n
)
(
1
)
=
(
-
1
)
n
+
1
n
!
ζ
(
n
+
1
)
ψ
(
n
)
(
1
2
)
=
(
-
1
)
n
+
1
n
!
(
2
n
+
1
-
1
)
ζ
(
n
+
1
)
ψ
′
(
n
+
1
2
)
=
1
2
π
2
-
4
∑
k
=
1
n
1
(
2
k
-
1
)
2
As
z
→
∞
in
|
ph
z
|
≤
π
-
δ
(
<
π
)
ψ
′
(
z
)
∼
1
z
+
1
2
z
2
+
∑
k
=
1
∞
B
2
k
z
2
k
+
1
