On my old fractional calculus approach to hyper-operations

[JmsNxn]

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...

As to your first point. I shared a lot of work at U of T; and they started calling these things hyper-operation chains (at least the people I talked to). There isn't anything published as of yet; but they've done quite a few things similarly to me. They never published, I presume because I have priority over these fractional calculus things; at least from their perspective. I kind of left the scene for a while; and they were upset I never published half the things they sort of knew about through me. It's why I've started publishing all over again; sort of like a code dump of everything I've done. Largely because some professors told me to.

They did do some stuff with,

\(
\alpha \uparrow^s z\\
\)

But I rarely see them (especially with covid right now); and I presume it's slow going. But they seemed confident my original formula for it 5 or 6 years ago is the correct one (but my original proof is incorrect):

\(
\alpha \uparrow^s z = \frac{d^{s-1}}{dw^{s-1}} \frac{d^{z-1}}{du^{z-1}} ||_{w=0}_{u=0} \sum_{n=0}^\infty \sum_{k=0}^\infty \alpha \uparrow^{n+1}(k+1) \frac{w^nu^k}{n!k!}\\
\)

They were also the ones who encouraged me to publish all this Infinite composition stuff. They were pretty shocked when I explained the residual theorem to them (just like you were ).

---

As to your second point; I would absolutely avoid talking about \( \theta \in (\pi/2,\pi) \) (the value \( \theta = \pi \) is out of the question too; because then it's bounded on \( \mathbb{C} \) and it's just constant). If you want to include sectors of this length; things can get a bit more complicated. In fact; it's a good amount trickier in these cases. So, I only play with \( \theta \in (0,\pi/2] \). Not that you can't use these cases; but if my memory serves me correct; the functional properties change a fair amount.

But yes, for every \( \theta \in (0,\pi) \) we have the correspondence \( \mathbb{E}_\theta \simeq \mathbb{S}_\theta \).

We absolutely have the restriction you are asking. Yes, we can view these as operators acting on restricted spaces and they're equivalent. I mean,

\( F \in \mathbb{E}_\theta \)


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

Has no dependency on \( \theta \) so the restriction is arbitrary.

Additionally;

\( f \in \mathbb{S}_\theta \)

Then,

\(
\Gamma(z) F(-z) = \int_0^\infty f(-y)y^{z-1}\,dy\\
\)

Which again, has no mention of \( \theta \). The variable only appears to describe the asymptotics of \( f \) and \( F \). Where for \( f \) it determines the size of the sector of its convergence. And for \( F \) it describes its possible growth type as \( \Im(z) \to \pm \infty \).

And yes, you can absolutely work with,

\(
\mathbb{E} = \bigcup_{\theta \in (0,\pi)} \mathbb{E}_\theta
\)

As I usually only care about \( \theta = \pi/2 \) or \( \theta < \pi/2 \); I don't pay much mind of that. Remember though, we do not want \( \theta = 0 \). This is no good. The most important part is that we have an open sector; when \( \theta = 0 \) we just have a line; and this will produce anomalies. Particularly; it'll screw things up when you want to make the correspondence; because you cannot really "pull out" any asymptotic data.

What you are doing with this union is much more similar to the classical treatment of Ramanujan's Master Theorem. I don't like this treatment; largely because it avoids explicitly stating how the differintegrated function is bounded. And it's very helpful to know how its bounded. If we consider this union, we're not being explicit about where the integral converges; and we're stuck only with the absolute knowledge that,

\(
\Gamma(z)F(-z) = \int_0^\infty f(-y)y^{z-1}\,dy\\
\)

And for some sector it works. This will produce problems when you want to do more advanced things in functional analysis with these things (which is moreso needed for the function \( \alpha \uparrow^s z \), or for defining a convolution, or for introducing the indefinite sum). But yes, you are correct.

---

As to your last point, I could never find any use for this thing. A long time ago I used to try and try to create a convergence factor. So that, for arbitrary \( F \), not necessarily in \( \mathbb{E} \), there exists some \( G \in \mathbb{E} \) such that \( F\cdot G \in \mathbb{E} \).  Then we would get,

\(
\Gamma(z)F(-z) = \frac{1}{G(-z)}\int_0^\infty h(-y)y^{z-1}\,dy\\
\)

But I could never find anything meaningful...

I sort of settled that there's no way to force a function to be in \( \mathbb{E} \); it just is or it isn't.

Regards, James