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

[MphLee]

@Gottfreid txh for you great answer

However, having 4 partial sequences, this might moreover reflect some effect in the question of negative \(z \) and imaginary \( z \). This might then be interesting in its own...

Can we derive from this regular behavior modulo 4 that modulo 4 and the coefficient themselves, even if they produce divergence, are following some kind of law, have meaning?
It is that that you are pointing to?

Well, the process is a bit different for such summations.

The most simple case is that of alternating geometric series, for instance \( f(x)= 1-x + x^2-x^3+... \). This series has radius of convergence \( \rho = 1 \) because for \( | x| <1 \) this converges (possibly veeeery slow ) and for \( |x| \ge 1 \) diverges (again possibly veeeery slow).   
On the other hand, the series can be understood to be result of \( f^*(x) = 1/(1+x) \) and there is no problem to insert then \( x=2 \) for instance and have the result \( f^*(2) = 1/3 \) . Perhaps this is some axiomatic anchor for all the more sophisticated summations of divergent series.           
Anyway, this is surely a "singular" observation and, at least since Euler, it has been used to extend the range of functions which can be summed to meaningful values, even if the powerseries for some \( x \) is divergent.         
Known names (with simple procedures) here are Hölder- and Cesaro summation, which apply evaluation of partial series up to some index \(n \), and average then that partial-series-values and see, whether this goes to some fixed limit. Iteration improves the "power": stronger diverging series can be summed.

However: only up to certain divergence criteria.  For instance, if the coefficients of some powerseries grow geometrically, neither Hölder- nor Cesaro sum (the latter: I think) do not converge - how many iterations you may apply. For such series the Euler-summation is sufficient, as well possibly iterated.  

For alternating factorial series, L. Euler found some evaluation-scheme, but the best known "(classical)" method for them is perhaps the Borel-summation.
This is remarkable, since \(f(x) = 0! - 1!x + 2!x^2 - 3!x^4 \pm \cdots \) has convergence-radius \( \rho =0 \): with no \(x\), being as small as you want, the series can converge. And still there is a meaningful evaluation to a finite value known!   

Oh, so I was not totally off... average of the truncations is something pretty intuitive that also a kid could come up with and similat to what I was suggesting (average of accumulation points of the subsequence of the truncations).

Anyways... it is a very rich zoo of summation methods...

There are many more, and more modern summation-methods known, but I couldn't specialize on this subject, and just fiddled with that homebrewn procedure of my own -often simply generalizations: to make the summation-procedure "stronger" while keeping its basic logic - methods to approximate meaningful values.        
It is a subject that I love much, and if interested I might prepare some zoom- or skype-meeting to show that at work using Pari/GP and my matrices... :-)

I'd be happy but I'm not sure I have the prerequisites and also the time needed to apply on this. Life is short and time is a tirant.. I hate this since it sounds as a great topic... and now that you described it many things popped to my head, things that are linked to... like zeta functions and combinatorics (generating functions-ology) and quantum theory stuff...

But again, no time... and I fear I'll never be able to catch up with all the prerequisite. It is already odyssey to try to make a little sense of dynamics and category theory as an autodidact... I'd need an university and a way to go there... damn... maybe when I'll be 70yo I'll get my phd. xD

Buttttt - most books are really difficult to read, not much paedagogic, exotic jargons, lacking redundancies etc, except for instance that of K. Knopp, that of G. Hardy and the like, all such "classical" texts. Modern texts about this, as far as I met them, are usually too hard to chew for me, so I've settled to dwell as amateur... :-)

Gottfried   

------
An, and a very lowlevel introduction into summation of divergent series maybe this one, however again: I've been rather newbie with this): https://go.helms-net.de/math/summation/pmatrix.pdf chap. 5

I'll try to skim Knopp in the free time.
Btw... chapter 5 is the "Online res(S)ources" part... and I suspect you are pointing me to the geman languange bibliography items...are you?