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

[bo198214]

I agree there are alternate ways to iterate \( e^x-1 \), there are at least three ways I know of from my own research.

Well often there are lots of different ways to do things, the question is rather whether the outcome is the same. And I think it is important to note that your approach also just produces regular iteration (in the sense of Szekeres 1958) and not some other exotic iteration.

Schroeder summations are not an efficient to iterate \( e^x-1 \). It requires 2312 summations in order to evaluate the tenth term.

But this is only the lesser problem. Usually one is not interested to have an iterate as a non-convergent powerseries but one want to have a computable real/complex function. E.g. can you draw the graph of the regular half iterate of e^x-1 (or that of e^(x/e) which is equivalent)? Currently Dmitrii and I working on a paper showing the complex plot of the Abel function, its inverse and some iterates of e^(x/e).

What they do is show that there is a combinatorial structure underlying all iterated functions, Schroeder's Fourth Problem http://www.research.att.com/~njas/sequences/A000311 . Also Schroeder summations are produced using Faà di Bruno's formula which is an example of a Hopf algebra which is important in several different areas of quantum field theory including renormalization. It is my hope that this might shine some light on how to show that our formulations of iterated functions and tetration are actually convergent.

I am always a bit skeptical about too philosophical approaches. I think in mathematics there should always be some result that is applicable and helpful in solving some problem.