Today, 08:59 AM
It is important to highlight that this is not a rigorous explanation. I am not going to show proofs in this post. This is just to show an extension of the exponential factorial.
The Exponential Factorial in this post is defined as:
\[1!_{2} = 1\]
\[x!_{2} = x^{(x-1)!_{2}}\]
At the start of my tetration research(roughly 7 months ago), I had succeeded in extending the exponential factorial using Kneser Tetration(Base \(e\)). Let Kneser tetration(Base \(e\)) be \(\operatorname{sexp}(x)\), and the super logarithm(Base \(e\)) \(\operatorname{slog}(x)\). We first define a value:
\[D = \lim_{n \to \infty} \operatorname{slog}(n!_{2})-n\]
This can be rewritten as:
\[D = \lim_{n \to \infty} \operatorname{slog}(\ln^{\circ (n)}(n!_{2}))\]
The limit does converge, which is obvious when written in a different form:
\[D = \operatorname{slog}\left(\ln\left(\ln\left(2\right)+\frac{\ln\left(\ln\left(3\right)+\frac{\ln\left(\ln\left(4\right)+\frac{\ln\left(\ln\left(5\right)+\frac{\ln\left(\ln\left(6\right)+...\right)}{4!_{2}}\right)}{3!_{2}}\right)}{2!_{2}}\right)}{1!_{2}}\right)\right)\]
\[\ln\left(\ln\left(2\right)+\frac{\ln\left(\ln\left(3\right)+\frac{\ln\left(\ln\left(4\right)+\frac{\ln\left(\ln\left(5\right)+\frac{\ln\left(\ln\left(6\right)+...\right)}{4!_{2}}\right)}{3!_{2}}\right)}{2!_{2}}\right)}{1!_{2}}\right) \approx\ -0.0613312423001\]
Using \(D\) we can extend the exponential factorial using Kneser Tetration(Base \(e\)) and the Super Logarithm(Base \(e\)):
\[x!_{2} = \lim_{k \to \infty} \log_{x+1}\left(\log_{x+2}\left(...\log_{x+k}\left(\operatorname{sexp}(x+k+D)\right)\right)\right))\]
I believe this is non-unique based on my numerical findings, however I have been unsuccessful to prove this is non-unique as well as to prove that it is unique. This extension does lead to a deep rabbit hole and an interesting phenomenon which does explain the NBLR(Nixon-Banach-Lambert-Raes) construction and other observations, which I have been researching for the past 7 months since I started my research on tetration, and I am currently writing a paper on it.
Also for the complex plane, I do not recommend using the extension directly. Instead take a taylor series at a point on the real axis above -1, and then evaluate at the complex point(depending on radius of convergence and all that).
I have approximated this extension in Desmos: https://www.desmos.com/calculator/jycrxkoslg
If one does have questions, please do ask!
The Exponential Factorial in this post is defined as:
\[1!_{2} = 1\]
\[x!_{2} = x^{(x-1)!_{2}}\]
At the start of my tetration research(roughly 7 months ago), I had succeeded in extending the exponential factorial using Kneser Tetration(Base \(e\)). Let Kneser tetration(Base \(e\)) be \(\operatorname{sexp}(x)\), and the super logarithm(Base \(e\)) \(\operatorname{slog}(x)\). We first define a value:
\[D = \lim_{n \to \infty} \operatorname{slog}(n!_{2})-n\]
This can be rewritten as:
\[D = \lim_{n \to \infty} \operatorname{slog}(\ln^{\circ (n)}(n!_{2}))\]
The limit does converge, which is obvious when written in a different form:
\[D = \operatorname{slog}\left(\ln\left(\ln\left(2\right)+\frac{\ln\left(\ln\left(3\right)+\frac{\ln\left(\ln\left(4\right)+\frac{\ln\left(\ln\left(5\right)+\frac{\ln\left(\ln\left(6\right)+...\right)}{4!_{2}}\right)}{3!_{2}}\right)}{2!_{2}}\right)}{1!_{2}}\right)\right)\]
\[\ln\left(\ln\left(2\right)+\frac{\ln\left(\ln\left(3\right)+\frac{\ln\left(\ln\left(4\right)+\frac{\ln\left(\ln\left(5\right)+\frac{\ln\left(\ln\left(6\right)+...\right)}{4!_{2}}\right)}{3!_{2}}\right)}{2!_{2}}\right)}{1!_{2}}\right) \approx\ -0.0613312423001\]
Using \(D\) we can extend the exponential factorial using Kneser Tetration(Base \(e\)) and the Super Logarithm(Base \(e\)):
\[x!_{2} = \lim_{k \to \infty} \log_{x+1}\left(\log_{x+2}\left(...\log_{x+k}\left(\operatorname{sexp}(x+k+D)\right)\right)\right))\]
I believe this is non-unique based on my numerical findings, however I have been unsuccessful to prove this is non-unique as well as to prove that it is unique. This extension does lead to a deep rabbit hole and an interesting phenomenon which does explain the NBLR(Nixon-Banach-Lambert-Raes) construction and other observations, which I have been researching for the past 7 months since I started my research on tetration, and I am currently writing a paper on it.
Also for the complex plane, I do not recommend using the extension directly. Instead take a taylor series at a point on the real axis above -1, and then evaluate at the complex point(depending on radius of convergence and all that).
I have approximated this extension in Desmos: https://www.desmos.com/calculator/jycrxkoslg
If one does have questions, please do ask!
