Intersections between this infinite power tower and a multifactorial

[Saracen Rue]

TL;DR

Find the smallest integer value for $k$ for which $x!^{k}$ has the largest number of intersections with ${}^{\infty}(!x)$, and state the values of $x$ and $y$ at each intersection point.

For those unaware of multifactorial notation, it should be noted that there are some common mistakes made when first being introduced to the notation. For example, $n! \neq (n!)!$ and $n! \neq (n!)! \neq (n!)! \neq ((n!)!)!$. Just to make sure we're all up to speed, here's a quick run down of how multifactorials are defined for integer values of $n$: $$n!=n(n-1)(n-2)(n-3)...(n-a)|\text{ } (n-a) > 0$$ $$n!=n(n-2)(n-4)(n-6)...(n-a)|\text{ } 2 \geq (n-a) > 0$$ $$n!=n(n-3)(n-6)(n-9)...(n-a)|\text{ } 3 \geq (n-a) > 0$$ $$\large \text{For a more generalized definition:}$$ $$n!^{k}= \left. \begin{cases}
1 & n=0\\
n & 1\leq n\leq k \\
n(n-k)(n-2k)(n-3k)...(n-a) & n>k
\end{cases}\right| k \geq (n-a) > 0$$ Continuing on from this, the next step is to define a function, $f_k(x)=x!^{k}$, such that $f_k$ is real and continuous for all values of $x$, which can be accomplished like so:
$$x!^{k}= \left. k^{\frac{x}{k}} \cdot \Gamma(1+\frac{x}{k}) {\prod_{j=1}^{k}}\left(\left(\frac{j \cdot k^{\frac{-j}{k}} }{\Gamma(1+ \frac{j}{k})}\right)^{\displaystyle \frac{1}{k} \sum_{n=1}^{k}\cos\left(\frac{2\pi n\left(x-j\right)}{k}\right)}\right)\right| x \in \mathbb{R}, k \in \mathbb{Z}^+$$ $$\large { \text{Where } !^{k} \text{ represents }k\text{ number of factorial symbols after }} x$$ Now for the second function in this question we need to talk about tetration and subfactorials. Focusing on the latter for now, let's look at the definition for a subfactorial (also known as derangement): $$!n=\text{subfactorial}(n)=\frac{\Gamma(n+1, -1)}{e}$$ $$\large \text{Where } \Gamma(a, x) \text{ is the incomplete gamma function}$$ This leaves us with the final topic of tetration. Tetration is a very useful tool and can be seen as another way of expressing Knuth's up-arrow notation. Consider the following: $$^{n}a=a\uparrow\uparrow n = \Large\underbrace{a^{a^{a^{.^{.^{.^a}}}}}}_{n \text{ times}}$$ Note: the notation can be kind of tricky here; "$n$ times" means that the total number of '$a$'s present is equal to $n$. For example: $$\Large ^{4}3={3^{3^{3^{3}}}}=10^{10^{12.56090264130034}}$$ When this is then applied to create an infinitely tall power tower we end up with this interesting relationship: $$\large ^{\infty}z=z^{z^{z^{.^{.^{.^z}}}}}=h(z)=\frac{-W(-\ln(z))}{\ln(z)}$$ $$\large \text{Where } W(z) \text{ is the Lambert W function}$$ It should be duly noted that $h(z)$ will converge if and only if $z$ falls in the domain $[e^{-e}, e^{\frac{1} {e}}]$. $$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$$ Now, with the definitions out of the way and hopefully everyone is on the same page, we can finally address the question; $$\large \text{Find the smallest integer value of } k \text{ for which}$$ $$\large f_k(x)=x!^{k} \text{ and }h(!x)=\frac{-W(-\ln(!x))}{\ln(!x)}$$ $$\large \text{have the most points of intersection for } x \geq 0$$ $$\large \text{and evaluate the coordinates for each point, correct to }3 \text{ decimal places}$$

 

[aheight]

Regarding this comment:

When this is then applied to create an infinitely tall power tower we end up with this interesting relationship: $$\large ^{\infty}z=z^{z^{z^{.^{.^{.^z}}}}}=h(z)=\frac{-W(-\ln(z))}{\ln(z)}$$ $$\large \text{Where } W(z) \text{ is the Lambert W function}$$ It should be duly noted that $h(z)$ will converge if and only if $z$ falls in the domain $[e^{-e}, e^{\frac{1} {e}}]$.

I believe it's more accurate to write:
Let:
$$y=\large^{\infty} z=z^{z^{z^{.^{.^{.}}}}}$$
and
$$D=\{z\in\mathbb{C}: y(z)\;\text{converges}\}$$
Then $y=z^y$ for $z\in D$ and we may write:
$$
y(z)=\frac{-\text{W}(-\log(z))}{\log(z)}; \;\; z\in D
$$
but the function
$$h(z)=-\frac{\text{W}(-\log(z))}{\log(z)}$$
is analytic (and multivalued) everywhere except at it's singular points.

Regarding this expression:

$$\large f_k(x)=x!^{k} \text{ and }h(!x)=\frac{-W(-\ln(!x))}{\ln(!x)}$$

If we plot the trajectory of $!x$ then when $x=1$ it's passing through the logarithmic branch point. Perhaps, though you could perturb the path around the origin and remain in analytic continuity on the same principal-valued sheet or even drop to the next sheet perturbing it in the opposite fashion and still remain analytically continuous. Not sure though how that would affect matters.