In mathematics , the K -function , typically denoted K (z ), is a generalization of the hyperfactorial to complex numbers , similar to the generalization of the factorial to the gamma function .
Definition [ edit ] 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 ) denotes the derivative of the Riemann zeta function , ζ (a ,z ) denotes the 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 .
Similar to the Bohr-Mollerup Theorem for the gamma function , the log K-function is the unique (up to an additive constant) eventually 2-convex solution to the equation Δ f ( x ) = x ln ( x ) {\displaystyle \Delta f(x)=x\ln(x)} where Δ {\displaystyle \Delta } is the forward difference operator.[3]
Properties [ edit ] 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 we have
∫ 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 in the OEIS ). References [ edit ] ^ Adamchik, Victor S. (1998), "PolyGamma Functions of Negative Order" , Journal of Computational and Applied Mathematics , 100 : 191–199, archived from the original on 2016-03-03 ^ Espinosa, Olivier; Moll, Victor Hugo , "A Generalized polygamma function" (PDF) , Integral Transforms and Special Functions , 15 (2): 101–115, archived (PDF) from the original on 2023-05-14 ^ "A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions: a Tutorial" (PDF) . Bitstream : 14. Archived (PDF) from the original on 2023-04-05. External links [ edit ]