01/07/2021, 05:44 PM
(This post was last modified: 01/07/2021, 08:03 PM by sheldonison.)
(01/07/2021, 09:53 AM)JmsNxn Wrote: 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?
- Sheldon