Bell matrices and Bell polynomials

[Gottfried]

I am looking at writing a follow up to
R. Aldrovandi and L. P. Freitas,
Continuous iteration of dynamical maps.
I want to understand the connection between Bell matrices and partial Bell polynomials. I think I am ready to work on the issue of reconciling Gottfried's work with mine.

 Nice! If I can I'd like to help. As far as I have seen, my ansatz is simply to use the Aldrovandi's ideas resp. Bell-matrices / partial Bell-polynomials, but because of the jungle of indexes and notations I'd given up to construct an explicite 1:1-representation of the (carleman-) matrices and the representations in Wikipedia or in Aldrovandi's article(s). For instance, my matrices (and their notation) are never thought as for multivariate polynomials in \(x_1,x_2,x_3,...,x_n\) but on series on one variable \(x\) only and a (infinite) set of constant coefficients, sloppily indicated as \( [a,b,c,...] \) or a bit more straight as \( a_0,a_1,a_2,...\)  (For the fractional iteration which includes a second parameter \( h \) for the iteration height there are mostly polynomials in \( h \) with order dependend in the exponent at \(x\) instead of constant coefficients \( [a,b,c,...] \), but this is not important at the moment).

Only very rarely I looked at polynomials in \( x \) (having only finitely many coefficients) but mostly at series. For the latter, the numerical examinations cannot simply assume truncation to matrices of finite size, but must explicitely check for effects which might occur, when the Carlemanmatrix assumes infinite size.  
A simple example for such effects is the problem of diagonalization of the Pascal-/Binomial-matrix, say \(P\), which provides the most basic operation on a "Vandermonde"-vector \([1,x,x^2,x^3,...] \to [1,x+1,(x+1)^2,(x+1)^3,...] \). For iterates, we can simply use the powers of \(P\). Even we can implement fractional iteration of this, since the fractional power of \( P \) can be expressed easily. But while the finite version of \(P\) can *not* be diagonalized, the infinite version can, and this might imply properties which are undiscussed when we only extrapolate from the finite version.  

I think it shall be impossible/unreadable-for-human-eye to express this latter, say "advanced", options with the given rich-index-language as in WP or Aldrovandi.

So, if I can, I'd like to help, but I likely shall not be able to navigate through the mentioned WP's or Aldrovandi's notational conventions...

Kind regards -

Gottfried