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

[Gottfried]

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?

Hi MphLee - 
 
thank you very much for considering this post!
And for the questions - no loss of time.

The reason for the separation is 99% because of appearence of the unseparated plot, resp. the list of coefficients. Looking at the coefficients-list, it is "obvious" (for me), that the 4-partioning reduces the apparent chaos in the list of coefficients (at the top of all is the effect of the periodicities of sign-changes - to smooth this a 4-partition is immediately useful). So: this is only heuristic.
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...

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.

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!   

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... :-)

Well, there are series, which are proven that they cannot be summed to some meaningful finite value that way at all, one of the examples is, if I recall correctly, a series with coefficients of this growth: \( f(x) = a + a^2 x + a^{2^2} x^2 + a^{2^3} x^3 + ... \text{with } \; a\gt 1 \) whether alternating or non-alternating signs are given. (But take me not at the word with this, this is from year old memory....).                         

So the statement of Erdös/Jabotinsky "no real fractional iterate for \( \exp(z)-1 \) " due to zero-radius of convergence has been a challenge to me, whether I can find some summation procedure for that series. (See my early more-or-less newbie discussion at https://go.helms-net.de/math/tetdocs/htm...ration.htm) Possibly there is indeed none; but my approximation suggests to me, that the growth of coefficients is somehow factorial ("hypergeometric") and then should be summable by Borel or similar procedures (see the formula for the \( a_k \) there are only factorial and geometric terms in it).  The best so far summation procedure can be seen in http://go.helms-net.de/math/tetdocs/Coef...ration.htm where I show a handful of summations of integer and of the half-integer powerseries (with \( x=1 \) , see pictures with blue background and yellow numbers)       

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

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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