logit coefficients growth pattern

[bo198214]

Quick question, I'm a little confused here.

Is this still guessing the asymptotics of a "half root" at a parabolic fixed point?

Or is it something different, (sorry just a tad confused).

We are looking at the growth of the coefficients of the logit, which shows quite the similar behaviour as the coefficients of the half root,
but speaks so to says about all *iterates* not only the half root. I didn't test it but I am super sure the 3rd root and 4th root, etc show the same behaviour - so the idea was to make it independent of the concrete iterative root, just dependent on the original function.
The logit is a good candidate for this - because the logit is analytic at the fixed point if and only if all regular iterates are analytic at the fixed point.
So I gave more examples for arbitrarily chosen parabolic functions where the logit shows a similar coefficient growth pattern as the logit coefficients of  \(e^x-1\), where the concrete growth behaviour depends on the \(x^2\)-coefficient of the function.
But then I also gave a counter example of a parabolic function where the logit is analytic (and hence all the regular iterates), i.e. the formal powerseries converges, i.e. the coefficients don't show that growth behaviour and don't need divergent summation.
And in the last post I just looked at the logit and it's relations for this counter-example from a non-powerseries pov and how you reconstruct the Abel function from the logit.

If this is happening elsewhere though; maybe Borel summation would be a valuable method of approaching fractional iteration?

By which we could get similar Euler expressions (Like how Euler analytically defines \(\sum_k (-1)^kk! z^k\)) of half iterates (and arbitrary iterates) using some kind of modified Laplace transform. All we would need is a bound like \(j_k = O(c^kk!)\).

Actually I wonder why Gottfried didnt post any results about the divergent summations he tried.
To get the left-side and the right-side iterates maybe one needs to apply two different divergent-summations