In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor Hugo Moll .[ 1]
It generalizes the polygamma function to negative and fractional order, but remains equal to it for integer positive orders.
The generalized polygamma function is defined as follows:
ψ
(
z
,
q
)
=
ζ
′
(
z
+
1
,
q
)
+
(
ψ
(
−
z
)
+
γ
)
ζ
(
z
+
1
,
q
)
Γ
(
−
z
)
{\displaystyle \psi (z,q)={\frac {\zeta '(z+1,q)+{\bigl (}\psi (-z)+\gamma {\bigr )}\zeta (z+1,q)}{\Gamma (-z)}}}
or alternatively,
ψ
(
z
,
q
)
=
e
−
γ
z
∂
∂
z
(
e
γ
z
ζ
(
z
+
1
,
q
)
Γ
(
−
z
)
)
,
{\displaystyle \psi (z,q)=e^{-\gamma z}{\frac {\partial }{\partial z}}\left(e^{\gamma z}{\frac {\zeta (z+1,q)}{\Gamma (-z)}}\right),}
where ψ (z )polygamma function and ζ (z ,q )Hurwitz zeta function .
The function is balanced, in that it satisfies the conditions
f
(
0
)
=
f
(
1
)
and
∫
0
1
f
(
x
)
d
x
=
0
{\displaystyle f(0)=f(1)\quad {\text{and}}\quad \int _{0}^{1}f(x)\,dx=0}
Several special functions can be expressed in terms of generalized polygamma function.
ψ
(
x
)
=
ψ
(
0
,
x
)
ψ
(
n
)
(
x
)
=
ψ
(
n
,
x
)
n
∈
N
Γ
(
x
)
=
exp
(
ψ
(
−
1
,
x
)
+
1
2
ln
2
π
)
ζ
(
z
,
q
)
=
Γ
(
1
−
z
)
ln
2
(
2
−
z
ψ
(
z
−
1
,
q
+
1
2
)
+
2
−
z
ψ
(
z
−
1
,
q
2
)
−
ψ
(
z
−
1
,
q
)
)
ζ
′
(
−
1
,
x
)
=
ψ
(
−
2
,
x
)
+
x
2
2
−
x
2
+
1
12
B
n
(
q
)
=
−
Γ
(
n
+
1
)
ln
2
(
2
n
−
1
ψ
(
−
n
,
q
+
1
2
)
+
2
n
−
1
ψ
(
−
n
,
q
2
)
−
ψ
(
−
n
,
q
)
)
{\displaystyle {\begin{aligned}\psi (x)&=\psi (0,x)\\\psi ^{(n)}(x)&=\psi (n,x)\qquad n\in \mathbb {N} \\\Gamma (x)&=\exp \left(\psi (-1,x)+{\tfrac {1}{2}}\ln 2\pi \right)\\\zeta (z,q)&={\frac {\Gamma (1-z)}{\ln 2}}\left(2^{-z}\psi \left(z-1,{\frac {q+1}{2}}\right)+2^{-z}\psi \left(z-1,{\frac {q}{2}}\right)-\psi (z-1,q)\right)\\\zeta '(-1,x)&=\psi (-2,x)+{\frac {x^{2}}{2}}-{\frac {x}{2}}+{\frac {1}{12}}\\B_{n}(q)&=-{\frac {\Gamma (n+1)}{\ln 2}}\left(2^{n-1}\psi \left(-n,{\frac {q+1}{2}}\right)+2^{n-1}\psi \left(-n,{\frac {q}{2}}\right)-\psi (-n,q)\right)\end{aligned}}}
where Bn (q )Bernoulli polynomials
K
(
z
)
=
A
exp
(
ψ
(
−
2
,
z
)
+
z
2
−
z
2
)
{\displaystyle K(z)=A\exp \left(\psi (-2,z)+{\frac {z^{2}-z}{2}}\right)}
where K (z )K -functionA is the Glaisher constant .
The balanced polygamma function can be expressed in a closed form at certain points (where A is the Glaisher constant and G is the Catalan constant ):
ψ
(
−
2
,
1
4
)
=
1
8
ln
A
+
G
4
π
ψ
(
−
2
,
1
2
)
=
1
2
ln
A
−
1
24
ln
2
ψ
(
−
3
,
1
2
)
=
3
ζ
(
3
)
32
π
2
ψ
(
−
2
,
1
)
=
−
ln
A
ψ
(
−
3
,
1
)
=
−
ζ
(
3
)
8
π
2
ψ
(
−
2
,
2
)
=
−
ln
A
−
1
ψ
(
−
3
,
2
)
=
−
ζ
(
3
)
8
π
2
−
3
4
{\displaystyle {\begin{aligned}\psi \left(-2,{\tfrac {1}{4}}\right)&={\tfrac {1}{8}}\ln A+{\frac {G}{4\pi }}&&\\\psi \left(-2,{\tfrac {1}{2}}\right)&={\tfrac {1}{2}}\ln A-{\tfrac {1}{24}}\ln 2&\\\psi \left(-3,{\tfrac {1}{2}}\right)&={\frac {3\zeta (3)}{32\pi ^{2}}}\\\psi (-2,1)&=-\ln A&\\\psi (-3,1)&={\frac {-\zeta (3)}{8\pi ^{2}}}\\\psi (-2,2)&=-\ln A-1&\\\psi (-3,2)&={\frac {-\zeta (3)}{8\pi ^{2}}}-{\tfrac {3}{4}}\\\end{aligned}}}
