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

[Gottfried]

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How are the derivates approximating exp(x) - 1 ?

I mean 

the 0.9999 iterate of exp(x) - 1 should give derivatives closer to those of exp(x) - 1 ... yet radius remains 0 ...

very confusing.

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what happens when we compare the taylor of the 0.499 iterate with that of the taylor of the 0.5001 iterate ?
Is the average that of the 0.5 iterate ??

Does the 0.6 iterate divided by the 0.4 iterate have a positive radius ??

still very confused.

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Hi Tommy -

this has been subject of some of my first ( ahem: most naive ;-) ) explorations at all.    
I have two types of views into it:

 a) first simply compare the numerical values of the coefficients of some fractional iteratives, up to index 64 or 128. It shows, that for fractional iteration heights the index \( k \) of coefficients from where on the coefficients begin to diverge (let's call this \( \kappa \) ), moves to a small value when \( h \) moves from integer to half-integer height. It allows the surprising formulation, that the iteration with integer \( h \) simply moves \( \kappa \) to infinity, or \( \lim_{\text{frac} (h) \to 0} \kappa = \infty \) .      
Some tabulations of that coefficients are here: tabulation1 and tabulation2 .

b) Another view in the characteristic of the growthrate of the coefficients occuring with fractional iteration heights are the following plots (below). Here I show curves for that tendencies, which I thought might even be more suggestive and perhaps helpful for to get a good idea how to formalize the growthrate depending on the distance of the fractional \( h \) from the integer values. See that plots at the end of this post.

Additionally - with this I thought to even have found a matrix-transformation for the powerseries-coefficients which produces a convergent series for the fractional iterate when also its argument \( x\) is transformed accordingly (using the matrix of Stirling numbers 2'nd kind). But I couldn't make this "fix" to a really safe conjecture/separate statement/essay. So this is still open and someone else might step in and see whether it is worth to explore this further. See the plots (there are also a couple of other bases than simply \( \exp(x)-1 \) tested)  (see also two plots of my essay at the end of this post)

So while both viewpoints gave interesting insights I had no further attempt (until 2016 with my growthrate conjecture recited in this thread) to formulate an upper-bound-conjecture for the growthrates of that coefficients - just observed, they might simply "hypergeometric" (as Euler christened this) - and "at least": meaning not knowing whether perhaps more.      
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Your keyword "Gevrey"-(class) might give the key to further leading reading here. Will Jagy in MO/MSE brought this into the discussions sometimes, but my attempt to read/understand about it had too little determination to really learn. So this is still open (and likely shall stay so, for my part).   

However, I would really like it, if my mentioned explorations could be "shoulders-of-dwarfs" ;-) on which some genius may take stand... Let's see...

Gottfried


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Here the attempt of a matrix-transformation which I gave the provisorical name "Stirling-transformation". The powerseries seem to transform to something that has nonzero radius of convergence: