Functional super-iteration and hierarchy of functional hyper-iterations

[bo198214]

Your operator was defined with my operator. Here is the definition that you gave:

I gave a recursive definition, your operator is defined with my and my is defined with yours. They are defined simulataneously by the equations

(1) \( (f \operatorname{It}_n g)(x) := (I_n^{g(x)} f)(x) \).
(2) \( (I_0^m f)(x) := f(m) \)
(3) \( I_n^1 f :=f \) and \( I_{n}^{m+1} f := f \operatorname{It}_{n-1} (I_n^m f). \)

Your I operator is basically the same as my It operator; the only differnce is the notation. Can you tell me exactly what is different about your definition of the hyper-
iteration rthan mine?

The difference is the declaration: Your operator It_n takes two functions and returns a function. My operator I_n takes a function and a number and returns a function.

[f It_n g](x) := [f It_n g(x)](x) - Let me clarify what it means.

Its absolutely clear what it means, however you see you make here the implicit distinction between It_n with a function as second argument (left side) and It_n with a number as second argument (right side).
To make the difference explicit rename It_n at the right side to I_n (and change the arrangement slightly). Then this exactly (1).

This equation can not be guessed! It needs to be written as part of the recursive definition!