matrix function like iteration without power series expansion

[Gottfried]

Guys! That I didnt see that before!

We have a very simple formula for computing the \( t \)-th iterate of an arbitrary function:

\( f^{\circ t} = \sum_{n=0}^\infty \left(t\\n\right) \sum_{k=0}^n \left(n\\k\right) (-1)^{n-k} f^{\circ k} \)

I want to add, that this expression exites me very much.

Just these days I was looking for alternatives (hopefully improvements) of our fractional iteration problem.
I was looking for two aspects

1. expression of the tetration by its own powertowers instead of powerseries
   
2. interpolation not done based on polynomial interpolation of the coefficients of the powerseries but based on an interpolation of gamma
   

The above formula provides both searched aspects in one shot.

The first is obvious; the idea is that only values are involved, which are provided by the function itself (so, for instance, possibly only of a relevant subset of complex numbers which are accessible by the function and its iterates itself/themselves - hmm, amateurish expressed; I hope the idea behind this can be understood anyway)

The second, because the fractional binomials use gamma of fractional parameters.

So - in my view - this is a big shot...

Gottfried