SuperFactorial

Fig.1. $y\!=\!x!$ and $y\!=\! \mathrm{SuperFactorial}(x)$ versus real $x$

SuperFactorial, or "superfactorial" is superfunction of factorial constructed at its fixed point 2.

The smallest integer larger than 2 (id est 3) is chosen as its value at zero, $\mathrm{SuperFactorial}(0)=3~$. Then,
$\mathrm{SuperFactorial}(1)$ $=$ $3!$ $=$ $6$,
$\mathrm{SuperFactorial}(2)=6!=720$,
$\mathrm{SuperFactorial}(3)=720!~$ and so on;

$\mathrm{SuperFactorial}(z)=\mathrm{Factorial}^z(3)$

Here, the upper index of a function indicates the number of iterations.

Superfactorial satisfies the transfer equation

$\mathrm{Factorial}(\mathrm{SuperFactorial}(z))=\mathrm{SuperFactorial}(z\!+\!1)$

SuperFactorial should not be confused with hyperfactorial , expressed with

$\displaystyle \mathrm{Hyperfactorial}(n)=\prod_{k=1}^{n}k^k$

for integer values of the argument and with relatively simple integral for the complex values.

Nest

Superfactorial can be expressed through the Mathematica built-in function Nest,

$\mathrm{SuperFactorial}[z]=\mathrm{Nest}[\mathrm{Factorial}, 3, z]$

For $z\in \mathbb N$, such an expression is recognized correctly.

$\mathrm{SuperFactorial}(z)$ is just $z$th iteration of Fatorial, evaluated at 3; so,

$\mathrm{SuperFactorial}(0)=3$

$\mathrm{SuperFactorial}(1)=\mathrm{Factorial}(\mathrm{SuperFactorial}(0))=\mathrm{Factorial}(3)=6$

$\mathrm{SuperFactorial}(2)=\mathrm{Factorial}^2(3)=\mathrm{Factorial}

\big( \mathrm{Factorial}(3)\big)=\mathrm{Factorial}(6)=6!=720$

$\mathrm{SuperFactorial}(3)=\mathrm{Factorial}^3(3)=\mathrm{Factorial}(720)=720!$

$\mathrm{SuperFactorial}(4)=\mathrm{Factorial}^4(3)=\mathrm{Factorial}(720!) ~$

and so on; however, the factorial of 720 is already too big and cannot be shown here as integer constant. Along the real axis, the SuperFactorial grows up faster than tetration does.

History

First, the superfunction of factorial is described in 2010 in the Moscow University Physics Bulletin . That publication defined the $\sqrt{!\,}$ as holomorphic function, giving sense to the logo of the Physics Department of that University. As a by–product, the SuperFunction of Factorial and that for some other special functions are considered there. Below, one of superfunctions of factorial is called SuperFactorial. The more detailed historic overview is presented in the article Square root of factorial.

Regular iteration

The straightforward application of the regular iteration to the solution $F$ of the transfer equation

$ (1)~ ~ ~ ~ ~ F(z\!+\!1) = \mathrm{Factorial}(F(z))$

is described below.

Aiming to build–up the growing function, the biggest among the real fixed points of factorial should be chosen as value at $-\infty$, so, search the solution in the following form:

$(2)~ ~ ~ ~ ~ F(z)=2+ a_1 \varepsilon + a_2 \varepsilon^2 +..$

where

$(3)~ ~ ~ ~ ~ \varepsilon=\exp(kz)$

It is convenient to set $a_1=1~$. Then other coefficients $a$ and constant $k$ can be determined substituting such a representation into the transfer equation (1):

$(4) ~ ~ ~ ~ ~ \mathrm{Factoral}(2+ \varepsilon + a_2 \varepsilon^2 +..) - (2+ K \varepsilon + K^2 a_2 \varepsilon^2 +..)=0$

where $K=\exp(k)$. The expansion of the left hand side of equation (1) into the power series with respect to $\varepsilon$ and equalizing to zero the coefficients gives the equation for $K$ and those for $a_2$, $a_3$, .. For example,

$\mathrm{Factorial}'(2)-K=0$

$\mathrm{Factorial}'(2) a_2+\mathrm{Factorial}(2)\frac{1}{2} a_1^2-K^2 a_2=0$

The derivatives of Factorial are known, in particular,

$\mathrm{Factorial}'(2)=3-2 \gamma$,

$\mathrm{Factorial}(2)= 2 - 6 \gamma + 2 \gamma^2 + \pi^2/3$

where $\gamma=-\mathrm{Factorial}'(0) \approx -0.5772156649015329$ is the Euler constant. Then,

$K=3-2\gamma~$ ; $~ ~ ~k=\ln(K)=\ln(3-2\!~\gamma) \approx 0.6127874523307$

$a_2=\frac{\pi^2+6\gamma^{2}-18\gamma+6}{12(3-5\gamma+2\gamma^{2})} \approx 0.79873183517243454$

and similar expressions for higher coefficients $a$.

The series (2) diverges, but the equation (1) allow to transfer the approximation from the range where it is valid (sufficient values of $-\Re(z)$) to other values and therefore get the required precision of evaluation of $F$.

In order to satisfy condition SuperFactorial(0)=3, it is sufficient to set

$\mathrm{SuperFactorial}(z)=F(x_0\!+\!z)$

where $~x_0\!\approx\! -0.91938596545217788~$ is solution of equation $~F(x_0)\!=\!3~$.

Properties of Superfactorial

Fig.2. Complex map of $f=\mathrm{SuperFactorial}(x+\mathrm i y)$ in the $x,y$ plane; levels $u=\Re(f)=$const and $v=\Im(f)=$const are drawn

The complex map of SuperFactorial is shown at the figure 2.

SuperFactorial is real-holomorphic, at lest in the range $\{z \in \mathbb C : \Re(z)\le 2\}$.

SuperFactorial is periodic function; the period

$T=\frac{2 \pi \mathrm i}{k} \approx 10.2534496811560279265772640691397~\mathrm i$

At non-negative integer values of the argument, SuperFactorial takes integer values; in particular, $\mathrm{SuperFactorial}(0)=3~$

Function $\tilde F(z)=\mathrm{SuperFactorial}\left(z+\frac{T}{2}\right)$ is also real–holomorphic; it smoothly decreases from value 2 at $-\infty$ to $1$ at $\infty$ along the real axis. Such a complementary superfunction typically exists at the consruction of a superfunction with regular iteration at a real fixed point.

Singularities

Fig.3. Zoom-in from Fig.2.

FuperFactorial is not entire function, over-vice, it would have negative integer values somewhere, which is not consistent with the transfer equation (1): Factorial at negative integers has poles. The unity translation form any point, where the SuperFactorial has negative integer value leads to a singularity.

The conclusion about the singularities is confirmed by the zoom-in from the comlpex map of the SuperFactorial. The figure indicates that the singularities of SuperFactorial are located within half-strips

$ z\in \mathbb C :$ $ \{ \Re(z) >\!2.7 ~,~ |\Im(z)\!+\!T n|<1 ~\mathrm{for~ some~} n\in \mathbb Z \}$

Outside these half-strips, the SuperFactorial exponentially approaches 2 as the real part of the argument goes to $-\!\infty$ and approaches unity as the real part of the argument goes to $+\!\infty$.

AbelFactorial

The inverse function of SuperFactorial is AbelFactorial; at least in some vicinity of the half-line along the real axis, the relations

$\mathrm{SuperFactorial}(

   \mathrm{AbelFactorial}(z))=z$

$\mathrm{AbelFactorial}(

   \mathrm{SuperFactorial}(z))=z$

The AbelFactorial satisfies the Abel equation

$\mathrm{Abelfactorial}( \mathrm{Factorial}(z))= \mathrm{AbelFactorial}(z)+1$

See the special article AbelFactorial for the details.

Together, the AbelFactorial and the SuperFactorial allow to express (and to evaluate) the iterations of Factorial in the following way:

$\mathrm{Factorial}^c(z)=\mathrm{SuperFactorial}\big(c + \mathrm{AbelFactorial}(z)\big)$

where number $c$ of iterations has no need to be integer; it can be even complex, and for some domain of $c,d,z\in C^3 \subseteq \mathbb C^3$, the relation

$\mathrm{Factorial}^c\big(\mathrm{Factorial}^d(z)\big)=\mathrm{Factorial}^{c+d}((z)$

as if takes place for the exponentiation. This justifies the notation square root of factorial for the half–iteration of Factorial, id est

$\sqrt{!~}~=~\mathrm{Factorial}^{1/2}$

In wide domain of values of $z$, the relation

$\sqrt{!~}~(\sqrt{!~}(z))~=~\mathrm{Factorial}(z)=z!$

See the special article square root of factorial for the details.

Existence and uniqueness

The existence and uniquness of a SuperFunction for a holomorphic fixed point is considered in . The regular iteration gives the explicit proof of existence, while the requirements on the behavior at zero and the asymptotic properties provide the uniqueness. However, there are many other fixed points, and each of them can be used to construct a superfunction of factorial, pretty different from the SuperFactorial described above.

The new superfunctions of Factorial can be constructed with periodic perturbation of its argument, let

$f(z)=\mathrm{SuperFactorial}(z+\zeta(z))$

where $\zeta$ is some hoomorphic function, periodic with period unity. Function $f$ satisfies the same transfer equation {1} as the SuperFactorial; however, the periodicity or $\zeta$ does not allow it to decay as the real part of the argument goes $-\infty$; as the result, neither the asymptotic behavior (2) of the SuperFactorial, nor its periodicity with period $T$ can be achieved. In this sense, with specification of the asymptotic behavior and the periodicity, the SuperFactorial is unique.

The SuperFactorial above seems to be the simplest among various SuperFunctions of Factorial, if one exclude functions that are identically equal to a constant. Other non-trivial superfunctions of Factorial show complicated behavior: and even the holomorphism in the left hand side of the complex plane cannot be achieved. Up to year 2011, a statement has status of conjecture, and the rigorous proof may be matter for the good outstanding research.

Algorithmic implementation of Superfactorial

C++ implementation

For the efficient evaluation of SuperFactorial, the regular iteration is suitable; the reasonable number of terms in the partial sum is of order of dozen. Then, within few tens of operations, one can get of order of 15 decimal digits, that is sufficient for the complex double implementation. The C++ code for the evaluation of SuperFactorial used to generate the figures is expected to be loaded, as a common component of the generators of the figures.

In the final version, the SuperFactorial is expected to be implemented as

complex<double> SuperFactorial(complex<double> z)

and its inverse function is expected to be implemented as

complex<double> AbelFactorial(complex<double> z)

The preliminary versions of the routines can be uploaded upon request; the final version will appear after to check the compatibility with other codes that are already loaded in TORI and in the Citizendium.

Mathematica implementation

In version of 2011, Mathematica already has name reserved for the iteration of functions, namely, Nest. The call of such a function may have form

Nest[f,a,z]

where $f$ indicates the name of the function, $z$ is number of iterations, and $a$ is initial value.

Then, $\mathrm{SuperFactorial}(z)$ can be expressed with

Nest[Factorial,3,z]

and the $c$th iteration of Factorial, id est, $\mathrm{Factorial}^c(z)$ could be written as

Nest[Factorial,z,c]

In the current version of Mathematica (year 2011), the Nest is implemented only for the case, when the last argument can be simplified to an constant , expressed with a natural number.

The upgrade of the implementation of Nest may include the table of SuperFunctions and corresponding Abel functions, their properties and ways of the efficient evaluation with required precision.

Conclusions

From the point of view of construction of superfunctions, there is nothing specific in the Factorial: the SuperFactorial is constructed in the same way as superfunction for $\exp_b$ for $1<b<\exp(1/\mathrm e)$, described in or that for the logistic operator .

SupeFactorial, together with its inverse function (id est, AbelFactorial) allows to define the $c$th iteration of factorial for non-integer values of $c$; in particilar, the square root of factorial (that was believed to have no sense ). appears as half-iteratiopn of factorial. This symbol of this function, id est, $\sqrt{!\,}$ is used as central part of logo of TORI.