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

[JmsNxn]

Yeah Gotfried at first sight about the a_k terms I thought about Borel summation, and I was also interested into divergent sums
I once also stucked in the summing-up terms d_n=O(exp(exp(n))) which is very much faster growing than O(n!) and any O(n!^k), I studied O(exp(exp(n))) because of the infinity q-pochhammer function \((q;q)_\infty\) which has nice expansions but O(q^n^2) cannot sum and even analytically extended to \(norm(q)>1\). I was so convinced about the contour integral method could sum diverge summations up, this may help idk
This may belong to the barren land of modern maths, but still I think it can guarantee a finite value as the sum, for example I used to extend the function
\(f(z)=\sum_{n\ge0}{z^{2^n}}\), firstly I used contour integration and it fits well, and it did also work outside \(\norm(q)\le1\), and also I used the property \(f(z^2)=f(z)-z\) to discover it's multivalued, and it might be a success?

Ironically, Leo. I wanted to approach this problem using work from Remmert's two part textbooks on complex analysis; which works on everything in analysis. There's a section on summing Taylor series on the boundary of the domain of holomorphy. And it describes (not siegel disks, but they call em something else, which are like siegel disks) how you can sum taylor series on the boundary of the domain of holomorphy. As odd as it sounds, your problem will behave way better than Gottfried's. I can solve your problem using the textbook, not Gottfrieds.

Your addition of complexity, actually satisfies a more regular structure. There is no divergent series! and it's easy to prove!

Gottfried's series is just on the boundary. And I mean it's the boundary of the boundary. And we should at least make a guess of \(O(k!^N)\) for some \(N > 1\). But it's definitely \(1+\delta >N > 1\), which can be better expressed as \(O(c^k k!)\) for this case in particular.

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This problem is a very difficult problem, exactly because it depends on being a boundary problem in more ways than one. And I think we're almost there.

I should add, that Remmert has a section on summing the very function you mention: \(\sum_{n=0}^\infty z^{2^n}\).