Borel summation

[Gottfried]

But isn't this quite \(f^{\circ 40}(h_{64}(f^{\circ -40}(x)))\), except that you Euler-summed \(h_{64}\) instead of just truncating the powerseries? This is what I meant - you actually don't need to care much about the power series, but you can get arbitrary precision via the iterations, i.e. just increase the 40, instead of to make efforts about the summation of the 64.

Yes, exactly. I had nothing better to contribute than this (what was already current practice in the forum), and actually I programmed my little tetration-modules in Pari/GP this way as likely everyone else.... (Either using the Schroeder-matrices/-functions or that what I called the "polynomial method" which simply used the diagonalization of truncated Carlemanmatrices - otherway staying in admiration of Sheldon's tool!).    

There has been no analysis about the growthrate of the coefficients in the powerseries for fractional iterations: may be they grow like \( c^{2^n} \) --- OMG - then the use as asymptotic series would have been misleading in the end! (see related discussion in MO https://mathoverflow.net/questions/19866...871#198871 , with example estimate/computation by Robert Israel and the impossibility of assigning a finite value to the series and a follow up by me https://mathoverflow.net/q/201098/7710 ).

Now we know, the series can be summed by Borel-summation; plus we find that the simple asymptotic evaluation produces the same -arbitrarily approximatable- values : so that's what I was missing all the time.  

I guess the MO-question  ( https://mathoverflow.net/q/4347 ) put together by Gil Kalai, and the following discussion, had left this point kept open in some way, has been only more-or-less explicite in this regard. And perhaps deserves now a new bullet point...   an additional answer composed by James...

Gottfried