Hey, Everyone; been a long time...

[JmsNxn]

The paper below is fairly involved, despite being only fifteen pages. It is self contained; but elements involve more general ideas formed in other work. Nonetheless, there forms a novel tetration. And everything you need to prove that is in this paper. I hope you all enjoy if you read it.

James,
I've been surprisingly busy this past year, but look forward to reading your paper.  As always, thanks for posting.  I have made zero progress on writing a paper proving convergence ....

I was quickly browsing your paper, and it sounds a little like Peter Walker's slog, which is constructed from the Abel function; this is from memory of Walker's approach ...
\( f(x)=\exp(x)-1;\;\;\;\alpha(f(x))=\alpha(x)+1 \)
The to generate a base "e" slog
\( \text{slog}_e(x)=\lim_{n \to \infty} \alpha(\exp^{[\circ n]}(x))-n \)
Then this is the slog_e(x) where we renormalize by a constant so that slog_e(1)=0.  
Walker proved his slog was infinitely differentiable, Henryk and I conjuctured it was nowhere analytic.  I believe there was a recent paper by Paulson claiming to rigorously prove Walker's slog was also nowhere analytic, but I am not convinced the proof was complete ....

If we used the superfunction instead \( \phi(x)= \alpha^{-1}(x) \) then is this the same as your approach?

HEY SHELDON! Long time no talk! I'm excited to discuss. Just because I'm being a little prideful, \( \phi(x) \) is NOT a superfunction. Not in any way shape or form. It's more like a superfunction, but with an exponential corrective term.

This is to mean that,

\(
\phi(s) = e^{\displaystyle s-1 + e^{s-2+e^{s-3....}}}
\)

So that,

\( \phi(s+1) = e^{s+\phi(s)} \)

(Believe it or not, this function is ENTIRE!)

Which I'm sure you can astutely note is NOT an inverted Abel function. The entire paper begins with constructing this function (which is the real novelty of the technique); and then, that old tried method that failed so many times, every time I read about it, finally works. Which is to say,

\(
\lim_{n\to\infty} \log \log \cdots (n\,\text{times})\cdots \log \phi(s+n) = e \uparrow \uparrow s + \omega
\)

(Which although it looks like Tommy's technique; or has wafts of Kouznetsov; it's absolutely more convenient to use \( \phi \).)

For some \( \omega \in \mathbb{R} \). The trick lies in using that,

\(
\log \log \cdots (n\,\text{times})\cdots \log \phi(s+n) = \phi(s) + \tau_n(s)
\)

For a corrective term \( \tau_n \), which kind of looks like \( s \). This sequence of \( \tau_n \) converge due to a clever use of Banach's Fixed Point Theorem. Which can be summarized as, \( \tau_1(s) =s \) and \( \tau_0(s) = 0 \); and they are generated through the recursion:

\(
\tau_{n+1}(s) = s + \log(1+\frac{\tau_{n}(s+1)}{\phi(s+1)})\\
|\tau_{n+1}(s) - \tau_n(s)| \le |\log(1+\frac{\tau_{n}(s+1)}{\phi(s+1)}) - \log(1+\frac{\tau_{n-1}(s+1)}{\phi(s+1)})|\\
\le \frac{1}{|\phi(s+1)|} |\tau_{n}(s+1) - \tau_{n-1}(s+1)|\\
\vdots\\
\le \Big(\prod_{k=1}^n \frac{1}{|\phi(s+k)|} \Big)|s+n|
\)

Where \( |s+n| = |\tau_1(s+n) - \tau_0(s+n)| \), by the initial conditions of the sequence. This product goes to zero geometrically, and thus the \( \tau_n \) converge. This of course, is much harder to do in reality (which is why I wrote the paper). We have to take supremum norms of these things in the complex plane; which requires understanding \( \phi \) very well in the complex plane.

So, to answer your question. As much as it looks like the Walker solution; as much as it borrows from the technique--I spent 3 years finding a function where it didn't suffer the same pitfalls. This is most definitely not a \( C^{\infty}(\mathbb{R}^+) \) solution. And I can prove that down to the \( \epsilon \) and \( \delta \). It is analytic.

Needless to say; I'm really excited for you to read it. If you have any questions I'm absolutely happy to answer. It is a little rough around the edges as a paper; but plug it in your calculator; and pay attention to my proofs of analycity. I think you'll be pleasantly surprised. It's a very different approach. But as hard as rock when it comes to the rigor involved.

Again, if you have ANY questions I'm happy to answer them.


Best regards, James

PS: I only came back to this forum, because 3 years ago, I said I'd only come back if I could construct a holomorphic tetration. So this paper has been 3 years in the making. Thank you, for giving it a chance.

PPS: Also when I uploaded it here it didn't render properly. I suggest using this link:

https://arxiv.org/pdf/2101.03021.pdf