Half-iterate exp(z)-1: hypothese on growth of coefficients

[MphLee]

There is a reason for subdividing the sequence of coefficients of the (diverging) series of the fractional iterate into 4 subsequences and not...say 3 or 6 or n subsequences?

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Read at risk of losing your time.

Let me start by saying that I'm totally a newbie with sums, and convergence, let alone evaluating divergent sums.
But the second question seems fascinating.

So I'll start with my childish feeling I got reading your post... if \(\sum d_kx^k\) diverges but somehow you can compute approximations, as naive as I can be, I understant that you are assigning to every \(x\) a value \({\rm Gottfried}(x)=G(x)\) in a way that up to some large \(N\), given by your computational power, we have \(|\sum_{k=0}^Nd_kx^k-G(x)|\leq \epsilon\) for some \(\epsilon\), i.e. the partial sum is not too much different from your approxximation.

As if there is something disturbing convergence, causing Baker's result, but the divergence is not so wild to make impossible for you to compute approximations.
So can we know if the partial sums are really wandering around the unknown real values like disturbed by some unknown "wind/disturbing force fiedl"?

So, a naive person like me, without background in computation nor summations, would ask: define the limit class of a sequence as \(\mathcal E(x_n)\) as the set of points that are the limits of some subsequence \(x_{n_k}\). The worst way \(x_n\) can diverge is when the limit class is empty.
Then I'd ask for \(\mathcal E(\sum d_nx^n)\) for each \(x\)... it doesnt matter it doesn't converge... but if it is nonempty for some \(x\in U\) then I'll look for a coherent way to chose from every \(\mathcal E_x\) a value in it.