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 There are multiple equivalent definitions of the K-function. The direct definition:
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].}
Definition via
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 , ζ ( s , q ) = ∑ k = 0 ∞ ( k + q ) − s {\displaystyle \zeta '(a,z)\ {\stackrel {\mathrm {def} }{=}}\ \left.{\frac {\partial \zeta (s,z)}{\partial s}}\right|_{s=a},\ \ \zeta (s,q)=\sum _{k=0}^{\infty }(k+q)^{-s}}
Definition via polygamma function:
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]}
Definition via balanced generalization of the polygamma function:
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.
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