On my old fractional calculus approach to hyper-operations

[MphLee]

The people at U of T called it a hyper-operator chain; that's not my terminology. I know it can be a tad confusing for this forum; but that's what they call it ; so I stuck with the terminology.

Yeah, but then I'm curious... how the hell did they know about hyperoperations? I'm pretty sure that serious mathematicians never talk about hyperoperations and the term hyperoperations is very niche and already used by hyperstructure theory (theory of groups with multivalued operation).
So if you tell me that they used that terminology for a reason... I'm pretty excited to hear more about that.
You know... 2015... 6 years googling things and the only persons that write that chain equation are you, Rubtsov and Romerio, 3/4 Tetration Forum's users and myself.

I guess your questions after that are about how I order the theorems. I guess it's just personal preference. You can always feed Ramanujan into Euler; that can be done even more generally than how I do it.

Mhh idk, I'll study this better. I had the impression that to ensure you could apply Euler to f you had to show FIRST that f=R[H] was in boldface E. I'm sure I have to read and understand better all those conditions (and probably go back to your old papers).

1.)
I'm a little confused by your first question;

I apologize... I'm sure I miss something crucial about convergence but I was thinking the following

I was asking if for EVERY \( \theta \in [0,\pi] \) we have  \( \mathbb{E}_\theta \simeq \mathbb{S}_\theta \)

The existence of that chain of inclusions is interesting... It should mean that we can extend \( {\mathfrak E}_w \) and \( {\mathfrak R} \) to a bigger domain. To be clear observe that if the origin of the complex plane is included and  \( \theta \le \kappa \) implies \( S_\theta \subseteq S_\kappa \) then  \( \displaystyle\bigcup_{\theta\in[0,\pi)}S_\theta ={\mathbb C}/(-\infty,0) \) and \( S_\pi=\displaystyle\bigcup_{\theta\in[0,\pi]}S_\theta ={\mathbb C} \)

From the monotone chain of inclusion also follows that for every \( \theta <\pi \) we have  \( \mathbb{E}_\theta \subset \mathbb{E}_\pi \) and  \( \mathbb{S}_\theta \subset \mathbb{S}_\pi \)

So you can't possibly mean that every theta is ok... maybe only for \( \theta \in [0,\pi) \)?

So the idea is the following.... if  \( \theta \le \kappa \) consider the two functions \( {\mathfrak E}^\theta_w:\mathbb{S}_\theta\to \mathbb{E}_\theta \) and \( {\mathfrak E}^\kappa_w:\mathbb{S}_\kappa\to \mathbb{E}_\kappa \) do we have that restricting \( {\mathfrak E}^\kappa_w \) to \( \mathbb{S}_\theta \) give us \( {\mathfrak E}^\theta_w \)?

In symbols \( {\mathfrak E}^\kappa_w|_{{\mathbb E}_\theta}={\mathfrak E}^\theta_w \)
Diagrammatically \( \mathbb{S}_\theta\overset{{\mathfrak E}^\theta_w}{\longrightarrow} \mathbb{E}_\theta\overset{\subseteq}{\longrightarrow} \mathbb{E}_\kappa \) is the same as \( \mathbb{S}_\theta \overset{\subseteq}{\longrightarrow} \mathbb{S}_\kappa\overset{{\mathfrak E}^\kappa_w}{\longrightarrow} \mathbb{E}_\kappa \)

If this condition works we can just work with spaces  \( {\mathbb S}:=\displaystyle\bigcup_{\theta\in[0,\pi)}{\mathbb S}_\theta \) and   \( {\mathbb E}:=\displaystyle\bigcup_{\theta\in[0,\pi)}{\mathbb E}_\theta \) because evey function in there satisfies your criterion for some \( \theta \), by definition.

2.)
This is a good question, that has a pretty deep answer.
[...]

\(
h(w) = \sum_{n=0}^\infty F(n)G(n) \frac{w^n}{n!}\\
\)

Woooa... that has to be important. I have some gut feeling that this is very important...
I'll keep it for myself now.... but I guess I have seen this somewhere before...