Is this THE equation for parabolic fix ?

[sheldonison]

What you have is not a Taylor series, so how did you find this expansion.

This series uses Jean Ecalle's FPS solution; there are many other posts on mathoverflow by Will Jagy and Henryk Trapman and Gottried Helms, about Jean Ecalle's parabolic solution.

Also this does not answer the op.

I know ways to get the Abel , but i forgot why getting to z=0 helps.
Probably perturbation theory.

It sounds like you're looking for the inverse of the parabolic solution for the Abel function, also developed around the fixed point, no? I haven't seen such an "inverse Abel function" formal power series; so it might be novel. The form might be similar to what was posted by Mick, on mathstack, but it would require a 1/x term to be the inverse of Ecalle's solution. Also, such an FPS would also likely have a zero radius of convergence for the same reasons that Ecalle's solution is an asymptotic series; which I briefly explained.

Btw for x + x^N the situation is different for Every N.

True; Will Jagy explains the general case in some of his posts. I just figured I would give the FPS series for iterating the parabolic case; \( f(z)=\exp(z)-1 \), to help you out. Scaling this FPS solution gives the solution for iterating \( \eta=\exp(1/e)\;\;\;f(z)=\eta^z\;\;\; \) which you mentioned earlier.

\( \alpha_\eta(z) = \alpha(\frac{z}{e}-1)\;\;\;\alpha_\eta(\eta^z)=\alpha_\eta(z)+1\;\;\; \) Abel function for \( f(z)=\eta^z \) in terms of the Abel function for \( f(z)=\exp(z)-1\;\;\; \) derived using \( \ln(\ln(\eta^{\eta^z})) \)