Some "Theorem" on the generalized superfunction

[MphLee]

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.

Can you expand on this?

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

Sure. I give you a latex version of an old answer I gave on MSE. Here you can find probably every reference on non-real ranks up to 2015. I have few more references but recent stuff.


  (2015) MphLee - Introduction to the non-integer ranks problem.pdf (Size: 391.07 KB / Downloads: 1,411)

If we find a proper definition of A+1 or (A^-1)+1 we can use binomial series to generalize its exponentiation
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

Maybe. This approach was recently discussed on this threads here (2021, JmsNxn's update of his 2015 approach) and here (2021, in the context of functors and operators).
in the past the problem of lack of linearity in the context of iterating operators using fractional calculus was observed also here (2015), here (2016), and here discussing the possible link with computability (2017).