Guys! That I didnt see that before!
We have a very simple formula for computing the
-th iterate of an arbitrary function:
 \sum_{k=0}^n \left(n\\k\right) (-1)^{n-k} f^{\circ k})
This series does not always converge but at least if
has an attracting fixed point reachable from 
then )
converges, because then )
is bounded, say |<M)
, then
|<M\sum_{n=0}^\infty \left(t\\n\right) \sum_{k=0}^n \left(n\\k\right)=M\sum_{n=0}^\infty \left(t\\n\right) 2^n = M3^t)
. I.e. the sum is absolutely convergent.
I bet my pants that this
is the regular iteration at the lower (attracting) fixed point in the case =b^x)
, 
.
Why is this matrix function like?
Well, if
has eigenvalues all smaller than 1 then, we dont need the Jordanization of , but the matrix power is also given as a convergent series
^t = \sum_{n=0}^\infty \left(t\\n\right) (A-I)^n=\sum_{n=0}^\infty \left(t\\n\right) \sum_{k=0}^n \left(n\\k\right) (-1)^{n-k} A^k)
.
If is the Carleman matrix of , then we just consider the first row in all those matrix powers and add them up.
Dont make the error however to assume that
^{\circ n}=\sum_{k=0}^n \left(n\\k\right) (-1)^k f^{\circ n})
This is not true. You can only do this with matrices as=AB+AC)
while mostly \neq f\circ g + f\circ h)
.
Notes:
We have a very simple formula for computing the
This series does not always converge but at least if
I bet my pants that this
Why is this matrix function like?
Well, if
If
Dont make the error however to assume that
This is not true. You can only do this with matrices as
Notes:
- Yes, this looks like the double binomial formula, note however that the formula should not be rearranged into the form
because this can lead to non-convergence. This is best seen with the formula
it must not be rearranged into the form
because 0 is a singularity for this function (
) and hence there is no power series development at 0.
- Perhaps Gottfried can jump in to provide summability in the divergent case
.
