09/07/2007, 02:15 PM

I think what you have done is similar what I describe now:

Given the exponentiation to base s: .

We know that it has the (lower) fixed point t with .

Let then

is a function with fixed point 0 and with .

Further it is known that a power series f with , and is conjugate to the linear function . As in our case and hence we apply it to our :

.

Now lets express this with power derivation matrices, let be the power derivation matrix of . Then we have:

.

We have the following correspondences

.

Now lets have a look at the structure of . We can decompose where is the lower triangular Pascal matrix given by . This is because the th row of the power derivation matrix of consists of the coefficients of . As we conclude

.

I think this completes the correspondences.

Unfortunately this decomposition works merely if the function under consideration has a fixed point. In so far it is very interesting that it also converges for the non-fixed point case with base .

Given the exponentiation to base s: .

We know that it has the (lower) fixed point t with .

Let then

is a function with fixed point 0 and with .

Further it is known that a power series f with , and is conjugate to the linear function . As in our case and hence we apply it to our :

.

Now lets express this with power derivation matrices, let be the power derivation matrix of . Then we have:

.

We have the following correspondences

.

Now lets have a look at the structure of . We can decompose where is the lower triangular Pascal matrix given by . This is because the th row of the power derivation matrix of consists of the coefficients of . As we conclude

.

I think this completes the correspondences.

Unfortunately this decomposition works merely if the function under consideration has a fixed point. In so far it is very interesting that it also converges for the non-fixed point case with base .