lpetrich
Contributor
Can one construct a sort of super factorial function? Yes.
Hyperfactorial and Hyperfactorial -- from Wolfram MathWorld
\( \displaystyle{ H(n) = \prod_{k=1}^n k^k } \)
It can be generalized to non-integer arguments like the factorial function:
K-function and K-Function -- from Wolfram MathWorld
\( H(n) = K(n+1) \)
with integral definition
\( \displaystyle{ K(z) = (2\pi)^{-(z-1)/2} \exp \left( \frac{z(z-1)}{2} + \int_0^{z-1} \log \Gamma(t+1) \, dt \right) } \)
A related funciton is the
Barnes G-function and Barnes G-Function -- from Wolfram MathWorld
For integer arg: \( \displaystyle{ G(n+2) = \prod_{k=1}^n k! } \) and \( \displaystyle{ G(n+1) = \frac{(n!)^n}{H(n)} } \)
For general arg: \( \displaystyle{ G(z) = \frac{\Gamma(z)^{z-1}}{K(z)} } \)
For general arg, it is given by an infinite product:
\( \displaystyle{ G(z+1) = (2\pi)^{z/2} \exp \left( - \frac{z + z^2(\gamma + 1)}{2} \right) \prod_{k=1}^\infty \left[ \left( 1 + \frac{z}{k} \right)^k \exp \left( \frac{z^2}{2k} - z \right) \right] } \)
Hyperfactorial and Hyperfactorial -- from Wolfram MathWorld\( \displaystyle{ H(n) = \prod_{k=1}^n k^k } \)
It can be generalized to non-integer arguments like the factorial function:
K-function and K-Function -- from Wolfram MathWorld\( H(n) = K(n+1) \)
with integral definition
\( \displaystyle{ K(z) = (2\pi)^{-(z-1)/2} \exp \left( \frac{z(z-1)}{2} + \int_0^{z-1} \log \Gamma(t+1) \, dt \right) } \)
A related funciton is the
Barnes G-function and Barnes G-Function -- from Wolfram MathWorldFor integer arg: \( \displaystyle{ G(n+2) = \prod_{k=1}^n k! } \) and \( \displaystyle{ G(n+1) = \frac{(n!)^n}{H(n)} } \)
For general arg: \( \displaystyle{ G(z) = \frac{\Gamma(z)^{z-1}}{K(z)} } \)
For general arg, it is given by an infinite product:
\( \displaystyle{ G(z+1) = (2\pi)^{z/2} \exp \left( - \frac{z + z^2(\gamma + 1)}{2} \right) \prod_{k=1}^\infty \left[ \left( 1 + \frac{z}{k} \right)^k \exp \left( \frac{z^2}{2k} - z \right) \right] } \)