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K-function

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For the k-function, see Bateman function.

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

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.}

It can also be given in closed form as

K ( z ) = exp ⁡ [ ζ ′ ( − 1 , z ) − ζ ′ ( − 1 ) ] {\displaystyle K(z)=\exp {\bigl }}

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}}

Another expression using the polygamma function is

K ( z ) = exp ⁡ [ ψ ( − 2 ) ( z ) + z 2 − z 2 − z 2 ln ⁡ 2 π ] {\displaystyle K(z)=\exp \left}

Or using the balanced generalization of the polygamma function:

K ( z ) = A exp ⁡ [ ψ ( − 2 , z ) + z 2 − z 2 ] {\displaystyle K(z)=A\exp \left}

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.

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 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+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).

Similar to the multiplication formula for the gamma function:

∏ j = 1 n − 1 Γ ( j n ) = ( 2 π ) n − 1 2 n − n 2 {\displaystyle \prod _{j=1}^{n-1}\Gamma \left({\frac {j}{n}}\right)=(2\pi )^{\frac {n-1}{2}}n^{-{\frac {n}{2}}}}

there exists a multiplication formula for the K-Function involving Glaisher's constant:

∏ j = 1 n − 1 K ( j n ) = A n 2 − 1 n n − 1 12 n e 1 − n 2 12 n {\displaystyle \prod _{j=1}^{n-1}K\left({\frac {j}{n}}\right)=A^{\frac {n^{2}-1}{n}}n^{-{\frac {1}{12n}}}e^{\frac {1-n^{2}}{12n}}}

References

  1. Adamchik, Victor S. (1998), "PolyGamma Functions of Negative Order", Journal of Computational and Applied Mathematics, 100 (2): 191–199, doi:10.1016/S0377-0427(98)00192-7, archived from the original on 2016-03-03
  2. Espinosa, Olivier; Moll, Victor Hugo (2004) , "A Generalized polygamma function" (PDF), Integral Transforms and Special Functions, 15 (2): 101–115, doi:10.1080/10652460310001600573, archived (PDF) from the original on 2023-05-14
  3. Marichal, Jean-Luc; Zenaïdi, Naïm (2024). "A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions: a Tutorial" (PDF). Bitstream. 98 (2): 455–481. arXiv:2207.12694. doi:10.1007/s00010-023-00968-9. Archived (PDF) from the original on 2023-04-05.
  4. Sondow, Jonathan; Hadjicostas, Petros (2006-10-16). "The generalized-Euler-constant function γ(z) and a generalization of Somos's quadratic recurrence constant". Journal of Mathematical Analysis and Applications. 332: 292–314. arXiv:math/0610499. doi:10.1016/j.jmaa.2006.09.081.

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