Matrix-method: compare use of different fixpoints

[bo198214]

I want your acknowledgement on this.

Well, you say it in the next sentence: let truncation aside. In all my considerations ...

Is this a confirmation?

Gottfried though I am not 100% clear about your my-oh-my method, it seems as if it does nothing more than to find the regular iteration at a fixed point, we discussed something similar already here.

The regular iteration of a function \( f \) with fixed point at 0 is given by
\( f^{\circ t} = \sigma^{-1}\circ \mu_{c^t} \circ \sigma \) with \( c=f'(0) \), \( \mu_d(x)=d\cdot x \) and \( \sigma \) being the principal Schroeder function.
If the fixed point is not at 0 but is at \( a \) then \( f=\tau_{-a}\circ \exp_b \circ \tau_a \) is a function with fixed point at 0 and \( f'(0)=c=\ln(a) \), where \( \tau_d(x)=d+x \). Putting this into the first equation we get:
\( \exp_b^{\circ t} = \tau_a\circ\sigma^{-1}\circ\mu_{{\ln(a)}^t}\circ\sigma\circ \tau_{-a} \).

But if we translate this into matrix notation by simply replacing \( \circ \) by the matrix multiplication and replacing each function by the corresponding Bell matrix (which is the transposed Carleman matrix) then we see a diagonalization of \( B_b \) because the Bell matrix (and also the Carleman matrix) of \( \mu_{\ln(a)}(x)=\ln(a)x \) is just your diagonal matrix \( {_dV}(\ln(a)) \)!
So it is nothing new, that we can diagonalize the untruncated \( B_b \) for each fixed point \( a \) with the diagonal matrix \( {_dV}(\ln(a)) \). As result we only get the plain old regular iteration at a fixed point.