Transseries, nest-series, and other exotic series representations for tetration

[Daniel]

The most general method I've developed for extending tetration is based on using a system of nested summations like you are talking about.

So you can compute a converging nested series for a fractional iterate of \( e^x-1 \)? This would be very interesting as it was shown by Baker and Écalle that the (ordinary) power series of the regular iterates of \( e^x-1 \) do not converge (except for integer iterates of course). I think though there is a paper of Écalle where he shows that they are Borel-summable despite.

I can compute a nested series for the fractional iterates of \( e^x-1 \), but I don't claim the series converges. I think the series is a formal power series. It is interesting to know that the series is Borel-summable.

Unfortunately its difficult (by a lack of theorems/propositions) to see on your site what you are actually able to do with your sums.

I developed Schroeder summations while looking for a tool that illuminates the combinatorial structure underlying iterated functions \( f^n(z) \) and their derivatives \( D^mf^n(z) \). So Schroeder summations are general, they are relevant to \( e^x-1 \), tetration, pentation and so on, just as long as the function is differentiable in the complex plane and has a fixed point.

Schroeder summations are consistent with what is known from complex dynamics, particularly the classification of fixed points. In fact the Schroeder summations can be used to derive and explain in detail the classification of fixed points including Schroeder's equation and Abel's equation and the conditions in which they can be used.