Some "Theorem" on the generalized superfunction

[Leo.W]

Thank you for the details. I really needed it!
...
bo198214, Jabotinsky's iterative logarithm, 2008

Indeed. I haven't been searching the oldest ref, not aware of someone did this great work so soon.
You can also show that if an iteration has RS model expression, you can't avoid a definition of Julia's equation, you can take notice at any Zf=gZ, take derivative and there's only one way to make the equation contains only Z' rather than Z&Z' or Z'&Z'', which means the whole RS definition is exactly a derivation from the Abel's equation. 

btw I wonder could you please introduce some ref about the continuous hyperoperators it'll be a great favor
if it's convenient for you

I had this quick thought (maybe coincident with early posts)
It's been 2 years since I concerned whether there lies a method to generalize an operator T to all of its powers
For instance, an ODE can be written as H(y)=0 where H is an operator, solving this ODE is exactly doing y=H^-1(0), I'm already aware that the method work for ODE sometimes and already showed in some essays.
Likewise, Iteration defined by the operator A and B, whose inverse
A^-1(f)=f(P(f^-1)) answers ''whose superfunction is f''
B^-1(f)=f^-1(P(f)) answers ''whose abel function is f''
where P is ''plus one'' operator, 
If we find a proper definition of A+1 or (A^-1)+1 we can use binomial series to generalize its exponentiations
I wonder if this method works, and what about on other operators, like d/dx? I maybe test fractional calculus some day soon.



Regards, Leo