In mathematics , the K -functionK (z ), is a generalization of the hyperfactorial to complex numbers , similar to the generalization of the factorial to the gamma function .
Definition [ ] Formally, the K -function is defined as
K
(
z
)
=
(
2
π
)
−
z
+
1
2
exp
[
(
z
2
)
+
∫
0
z
−
1
ln
Γ
(
t
+
1
)
d
t
]
.
{\displaystyle K(z)=(2\pi )^{-{\frac {z+1}{2}}}\exp \left[{\binom {z}{2}}+\int _{0}^{z-1}\ln \Gamma (t+1)\,dt\right].}
It can also be given in closed form as
K
(
z
)
=
exp
[
ζ
′
(
−
1
,
z
)
−
ζ
′
(
−
1
)
]
{\displaystyle K(z)=\exp {\bigl [}\zeta '(-1,z)-\zeta '(-1){\bigr ]}}
where ζ ′(z )derivative of the Riemann zeta function , ζ (a ,z )Hurwitz zeta function and
ζ
′
(
a
,
z
)
=
d
e
f
∂
ζ
(
s
,
z
)
∂
s
|
s
=
a
.
{\displaystyle \zeta '(a,z)\ {\stackrel {\mathrm {def} }{=}}\ \left.{\frac {\partial \zeta (s,z)}{\partial s}}\right|_{s=a}.}
Another expression using the polygamma function is[1]
K
(
z
)
=
exp
[
ψ
(
−
2
)
(
z
)
+
z
2
−
z
2
−
z
2
ln
2
π
]
{\displaystyle K(z)=\exp \left[\psi ^{(-2)}(z)+{\frac {z^{2}-z}{2}}-{\frac {z}{2}}\ln 2\pi \right]}
Or using the balanced generalization of the polygamma function :[2]
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 A is the Glaisher constant .
Properties [ ] It can be shown that for α > 0
∫
α
α
+
1
ln
K
(
x
)
d
x
−
∫
0
1
ln
K
(
x
)
d
x
=
1
2
α
2
(
ln
α
−
1
2
)
{\displaystyle \int _{\alpha }^{\alpha +1}\ln K(x)\,dx-\int _{0}^{1}\ln K(x)\,dx={\tfrac {1}{2}}\alpha ^{2}\left(\ln \alpha -{\tfrac {1}{2}}\right)}
This can be shown by defining a function f such that:
f
(
α
)
=
∫
α
α
+
1
ln
K
(
x
)
d
x
{\displaystyle f(\alpha )=\int _{\alpha }^{\alpha +1}\ln K(x)\,dx}
Differentiating this identity now with respect to α yields:
f
′
(
α
)
=
ln
K
(
α
+
1
)
−
ln
K
(
α
)
{\displaystyle f'(\alpha )=\ln K(\alpha +1)-\ln K(\alpha )}
Applying the logarithm rule we get
f
′
(
α
)
=
ln
K
(
α
+
1
)
K
(
α
)
{\displaystyle f'(\alpha )=\ln {\frac {K(\alpha +1)}{K(\alpha )}}}
By the definition of the K -function we write
f
′
(
α
)
=
α
ln
α
{\displaystyle f'(\alpha )=\alpha \ln \alpha }
And so
f
(
α
)
=
1
2
α
2
(
ln
α
−
1
2
)
+
C
{\displaystyle f(\alpha )={\tfrac {1}{2}}\alpha ^{2}\left(\ln \alpha -{\tfrac {1}{2}}\right)+C}
Setting α = 0
∫
0
1
ln
K
(
x
)
d
x
=
lim
t
→
0
[
1
2
t
2
(
ln
t
−
1
2
)
]
+
C
=
C
{\displaystyle \int _{0}^{1}\ln K(x)\,dx=\lim _{t\rightarrow 0}\left[{\tfrac {1}{2}}t^{2}\left(\ln t-{\tfrac {1}{2}}\right)\right]+C\ =C}
Now one can deduce the identity above.
The K -function is closely related to the gamma function and the Barnes G -function ; for natural numbers n , we have
K
(
n
)
=
(
Γ
(
n
)
)
n
−
1
G
(
n
)
.
{\displaystyle K(n)={\frac {{\bigl (}\Gamma (n){\bigr )}^{n-1}}{G(n)}}.}
More prosaically, one may write
K
(
n
+
1
)
=
1
1
⋅
2
2
⋅
3
3
⋯
n
n
.
{\displaystyle K(n+1)=1^{1}\cdot 2^{2}\cdot 3^{3}\cdots n^{n}.}
The first values are
1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... (sequence A002109 OEIS ). References [ ] External links [ ] ![{\displaystyle K(z)=(2\pi )^{-{\frac {z+1}{2}}}\exp \left[{\binom {z}{2}}+\int _{0}^{z-1}\ln \Gamma (t+1)\,dt\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f09a7505a55a212e2164ba9f423a48d1ffa227f6)
![{\displaystyle K(z)=\exp {\bigl [}\zeta '(-1,z)-\zeta '(-1){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12668025758db5c6b204ac5b1f43c1e8e05785a5)
![{\displaystyle K(z)=\exp \left[\psi ^{(-2)}(z)+{\frac {z^{2}-z}{2}}-{\frac {z}{2}}\ln 2\pi \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/527eb0c1daf22e5bb0b92ff2ff331e64b68f597e)
![{\displaystyle K(z)=A\exp \left[\psi (-2,z)+{\frac {z^{2}-z}{2}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e5a0771140cd752c427198127287d5cc8b0466)
![{\displaystyle \int _{0}^{1}\ln K(x)\,dx=\lim _{t\rightarrow 0}\left[{\tfrac {1}{2}}t^{2}\left(\ln t-{\tfrac {1}{2}}\right)\right]+C\ =C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e8c9ffa5aadcb54c1d1df9e5a769e78d7fa08a)
