Guys! That I didnt see that before!

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

This series does not always converge but at least if has an attracting fixed point reachable from then converges, because then is bounded, say , then

. 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 , .

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

.

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

This is not true. You can only do this with matrices as while mostly .

Notes:

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

This series does not always converge but at least if has an attracting fixed point reachable from then converges, because then is bounded, say , then

. 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 , .

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

.

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

This is not true. You can only do this with matrices as while mostly .

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 .