07/19/2022, 08:11 PM
(This post was last modified: 07/22/2022, 05:59 PM by Gottfried.

*Edit Reason: final update of pictures, removed caveats*)
Update 22.7.22: i replaced the pictures by a more "pedagogical" turn and took the original pictures to the end. Hopefully made the text accordingly clearer

Hi,

this is a bit older, but has not yet been discussed anywhere - I moreover don't think, the today's MSE or MO would like that question...

- - - - - - -

I.N. Baker has proved in the 1950'ies that the powerseries of any fractional iterates of \( f(z) = \exp(z)-1 \) have convergence radius \( \rho = 0\).

From this, for instance, Erdös/Jabotinsky took the stance: "no real fractional iterate of \( f(z) \) exists". (Citation-sources at my homepage, subpage "tetdocs")

Well, older members here might remember that I took procedures of divergent summation to actually compute approximations, for instance for the power series for the half-iterate \( g(g(z))=f(z) \quad \) ("regular iteration") - - - not to mention the option to look at the powerseries as an asymptotic one and take by this the partial sum up to best approximation....

The use of that divergent summation procedures was always due to inspection of partial sums up to 32,64,128 or 256 coefficients, and also handwoven (well, mostly very good heuristics).

To have a chance at all to make this more precise, ...

.... I tried to first find some estimate of the growthrate of the coefficients in \( g(z) \).

First see the coefficients \( g_k \) for \( k=0..45 \)

The initial coefficients are all small in absolute values, and short inspections of that problem might be misleading about the overall tendency.

Now look at the coefficients for \( k=0..1023 \) . I made an \( \sinh°^{-1}() \)-scaling for the y-axis of the picture to have a milder growthrate and much better overview of the drawn curves:

It is clearly a hypergeometrical growth in the sequence of coefficients, so of course this series should -by inspection of this 1024 first coefficients only should have convergenceradius zero - but it is not clear of this is all what can be said.

Sometime in 2016 I found a very suggestive good estimate: the curve of the coefficients \( g_k \) at index \( k \) multiplied with my reciprocal guess-function \( A(k) \) is extremely clearly bounded by a constant value of about \( \pm 10.01 \) .

- - - - - - - - - - - - - -

So since the limiting guess-function \(A(k) \) has a factorial in \( k \) in the numerator, Eulerian summation cannot be applied, and because there is nothing else that makes the growthrate more extreme, a Borel-summation (and likely my experimental Nörlund-summation) should indeed be sufficient to assign finite real values to the evaluation of \( g(z) \).

If this can be shown to be correct/appropriate, then P. Erdös'/E. Jabotinsky's unconditioned verdict seems to be overwritten.

Thus there are two aspects that I'd like to be solved (at any time in future...) :

I don't expect an answer to question 2. here in the forum, since divergent summation is a complete different field (and I've only peeked into it, at least finding some operationable versions).

But perhaps for question 1. someone might have an idea.

- - - - - - - -

Even if not, we might let this post hang around for possibly later ideas/forum visitors engagement. After tinkering with it I did no more expect MO being an appropriate place for questions of that style.

Gottfried

Appendix:

Here is the picture where moreover I document the scaled coefficients in 4 separate curves according to \( k \pmod 4 \). Maybe that four curves -or better: the four functions defined by the 4 "multi-section-series" - have any valuable properties on their own and might be analyzed using Fourier-analysis or else. See the image:

When the coefficients of second and third of the multi-section-series change their sign, then the two new common, sinusoidal curves even improve their shape ...

Hi,

this is a bit older, but has not yet been discussed anywhere - I moreover don't think, the today's MSE or MO would like that question...

- - - - - - -

I.N. Baker has proved in the 1950'ies that the powerseries of any fractional iterates of \( f(z) = \exp(z)-1 \) have convergence radius \( \rho = 0\).

From this, for instance, Erdös/Jabotinsky took the stance: "no real fractional iterate of \( f(z) \) exists". (Citation-sources at my homepage, subpage "tetdocs")

Well, older members here might remember that I took procedures of divergent summation to actually compute approximations, for instance for the power series for the half-iterate \( g(g(z))=f(z) \quad \) ("regular iteration") - - - not to mention the option to look at the powerseries as an asymptotic one and take by this the partial sum up to best approximation....

The use of that divergent summation procedures was always due to inspection of partial sums up to 32,64,128 or 256 coefficients, and also handwoven (well, mostly very good heuristics).

To have a chance at all to make this more precise, ...

.... I tried to first find some estimate of the growthrate of the coefficients in \( g(z) \).

First see the coefficients \( g_k \) for \( k=0..45 \)

The initial coefficients are all small in absolute values, and short inspections of that problem might be misleading about the overall tendency.

Now look at the coefficients for \( k=0..1023 \) . I made an \( \sinh°^{-1}() \)-scaling for the y-axis of the picture to have a milder growthrate and much better overview of the drawn curves:

It is clearly a hypergeometrical growth in the sequence of coefficients, so of course this series should -by inspection of this 1024 first coefficients only should have convergenceradius zero - but it is not clear of this is all what can be said.

Sometime in 2016 I found a very suggestive good estimate: the curve of the coefficients \( g_k \) at index \( k \) multiplied with my reciprocal guess-function \( A(k) \) is extremely clearly bounded by a constant value of about \( \pm 10.01 \) .

- - - - - - - - - - - - - -

So since the limiting guess-function \(A(k) \) has a factorial in \( k \) in the numerator, Eulerian summation cannot be applied, and because there is nothing else that makes the growthrate more extreme, a Borel-summation (and likely my experimental Nörlund-summation) should indeed be sufficient to assign finite real values to the evaluation of \( g(z) \).

If this can be shown to be correct/appropriate, then P. Erdös'/E. Jabotinsky's unconditioned verdict seems to be overwritten.

Thus there are two aspects that I'd like to be solved (at any time in future...) :

- is that bounding function "good"?

- What type of divergent-summation-procedure can be expected to sum that (strongly) divergent sum? (I simply used some Nörlund-like procedure based on heuristics & inspection)

I don't expect an answer to question 2. here in the forum, since divergent summation is a complete different field (and I've only peeked into it, at least finding some operationable versions).

But perhaps for question 1. someone might have an idea.

- - - - - - - -

Even if not, we might let this post hang around for possibly later ideas/forum visitors engagement. After tinkering with it I did no more expect MO being an appropriate place for questions of that style.

Gottfried

Appendix:

Here is the picture where moreover I document the scaled coefficients in 4 separate curves according to \( k \pmod 4 \). Maybe that four curves -or better: the four functions defined by the 4 "multi-section-series" - have any valuable properties on their own and might be analyzed using Fourier-analysis or else. See the image:

When the coefficients of second and third of the multi-section-series change their sign, then the two new common, sinusoidal curves even improve their shape ...

Gottfried Helms, Kassel