Superfactorial
In the Wolfram notations [1], Superfactorial appears as
\[ \mathrm{Superfactorial}(z) = \underbrace{ { z!^{.^{.^{z!}}} } }_{z!} = \mathrm{exp}_{z!}^{\ z!}(z!) = \mathrm{tet}_{z!}(z!) \]
In such a way, this Superfactorial is expressed through the tetration (4th ackermann) and the Factorial functions.
To year 2025, no specific property of such a Superfactorial is found that would not follow from properties of tetration and factorial; no any specific application for this function is found.
Confusion
This Superfactorial[1] should not be confused with SuperFactorial (SuFac) defined as Superfunction of Factorial [2][3][4], \[ \mathrm{SuFac}(z+1)=\mathrm{Factorial}(\mathrm{SuFac}(z)) \]
Here, Factorial plays role of the Transfer function and SuFac appears as its Superfunction.
The identifiers of the two functions,
SuperFactorial and
Superfactorial
differ with only single bit.
This may cause confusion, especially if an operational system confuses the Uppercase Letters
with their lowercase analogies.
Conclusion
The SuperFactorial SuFac and its inverse function AuFac\(=\mathrm{SuFac}^{-1}\) allow to express
the non-integer iterate of Factorial and, in particular, the square root of factorial (iterate half of factorial) [2][3][4].
No similar application for function Superfactorial[1] is found. For this reason, following the First of the TORI axioms, properties of Superfactorial are not considered in TORI.
References
- ↑ 1.0 1.1 1.2 https://mathworld.wolfram.com/Superfactorial.html The superfactorial of n is defined by Pickover (1995) as n$=n!^(n!^(·^(·^(·^(n!)))))_()_(n!). (1) // The first two values are 1 and 4, but subsequently grow so rapidly that 3$ already has a huge number of digits. ..
- ↑ 2.0 2.1
https://link.springer.com/article/10.3103/S0027134910010029
https://mizugadro.mydns.jp/PAPERS/2010superfae.pdf (Englsih version), https://mizugadro.mydns.jp/PAPERS/2010superfar.pdf (Russian version): D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12. (Russian version: p.8-14) - ↑ 3.0 3.1 https://mizugadro.mydns.jp/BOOK/202.pdf Дмитрий Кузнецов. Суперфункции. Lambert Academic Publishing, 2014
- ↑ 4.0 4.1 https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020
Keywords
«[[]]», «AbelFactorial», «Abelfunction», «Confusion», «Mathematica», «SuperFactorial», «Superfactorial», «Superfunction», «Superfunctions»,