Wed Mar 13 11:52:32 EDT 2013
Composition of series expansion
Is it actually possible to write this in a finite way? E.g.
exp(exp(x)) = sum_n 1/n! (sum_m 1/m! x^m) ^n
what is the coefficient of 1, x, x^2, ... ?
Hmm... I'm missing an important point here. The trouble is that this
is doubly infinite, so when doing this numerically, it is necessary to
add a termination criterion.
Only in the first coefficient though, meaning f(0), as it will show up
as an infinite sum. All other sums will be finite.
Finding the contribution for x^n depends on all possible splits of the
factorization of n, but is finite.
Another problem: the infinite sum might not even converge!
Actually, this is mentioned here on composition g(f(X))
A point here is that this operation is only valid when f(X) has no
constant term, so that the series for g(f(X)) converges in the
topology of R[[X]]. In other words, each cn depends on only a finite
number of coefficients of f(X) and g(X).