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

[Gottfried]

Okay, so I'm going to go out on a limb, and assume you mean that:

\[
f(z) = \exp(z) -1\\
\]
is expanded about \(z =0\), right?

I do have a long history of working with Divergent sums and Umbral Calculus and sum and series speed ups. So I'm very very happy to look into this. But I'm confused by what the bounding function is? I apologize, but I'm a tad confused.

I'd suggest an abel summation personally. And I'll explain how to do it. (...)

Hmm, I think there must be an error somewhere; for the powerseries for fractional iteration of \(f(z)=\exp(z)-1 \) the convergence-radius is not a small \( \delta \)-environment around zero, but exactly zero - that's not my discovery but that of I.N.Baker (and I could not yet understand fully his proof); so I think an Abel-summation should not work.    

I learned the term "limiting a divergent series" or "bounding a divergent series" from the handbook of Konrad Knopp; there he described this as a prerequisite to find an appropriate/powerfule-enough summation-procedure at all - and the growthrate of the coefficients of \( f°^h(z) \) must be estimated by his argument.   
That was the problem that I set myself up to solve: find a function \( A(k) \) (where \(k \) indicates the index of the coefficients in \( f°^h(z) \) ) which is always greater than the \( k \)'th coefficient in \( f°^h(z) \).   
After this - decide for the appropriate summation-technique.  If \( A(k) \) has "geometrical" growth (with \(k \) ) then Euler summation is appropriate/powerful enough, if it has "hypergeometric" growth then Borel-, or other powerful summation techniques are needed.           
This is a (sloppy) paraphrase of Knopp's explanations in chap 13 and 14 - and he didn't (if I recall it correctly) show any summation "stronger" than the Borel-summation; so when we possibly have coefficients in \( f°^h(z) \) growing more than factorially with their index, then there is no classical summation in his book to manage this (except perhaps his chapter about the Euler-MacLaurin summation and the use of asymptotic series, the latter which has explicitely been applied in this forum ... ).                

The function \( A(k) \) that I describe here has factorially growth (plus some geometric growth component) so any \( k \)'th coefficient in \( f°^h(z) \) is inside the bounds given by \( A(k) \) . What I've simply documented in my picture are the values \( a_k \) (and their structural expression) . Then -to have a sanity check- the quotient of the k'th coefficient in \( f°^h(z) \) with \( a_k \). The amazing thing in this is, that my proposal for the \(a_k\) seem lead to an upper limit in the same way as a sinus-curve has one.                 

Well - maybe this can be expressed far better, or even simpler. The literature I had was only Euler, Hardy, Knopp  and then some diverse articles, but no coherent or modern course...  so don't mind to correct me if I've got something basically wrong, I'd like it much if I could settle my experimental results into appropriate form someday :-) ...

Gottfried