Discussing fractional iterates of \(f(z) = e^z-1\)

[JmsNxn]

It seems to me the target of your paper is= "experts of the field".

It very much is, in this form. I plan to flesh out everything much more though, and give the "dummy's version", lol. This is still a bit cliff notes version.

I am also writing this because I loathe matrices, but I love Helms' and Trappman's analysis. But where you see infinite matricies, you also see integral transforms. So I'm trying to develop the integral transforms which are the exact same thing that Helms' and Trappman do on the hardy space (though they never mention hardy space). Quite literally it's the difference between Schrodinger's work, or Heisenberg's work. Integrals acting on a Hilbert space--or, Infinite Matrices acting on an infinite vector space \(\{1,z,z^2,z^3,...\}\).

I just want to translate it to integrals. And again, argue for the formula:

\[
f(z) = e^{z}-1\\
\]

\[
f^{\circ s}(z) = \frac{d^{s-1}}{dx^{s-1}}\Big{|}_{x=0} \sum_{n=0}^\infty f^{\circ n+1}(z)\frac{x^n}{n!}\\
\]

And that, the fractional calculus approach, produces the same result as Gottfried and Helms. I'm very excited  

I hope you're doing well MphLee! Good luck on all your work. I don't doubt you're possessed by these problems. I'm super excited to read your work!