08/11/2007, 11:42 PM

Daniel Wrote:exp(x)-1 definately has non-integer iterated; see heirarchies of height 1/2 at http://tetration.org/Combinatorics/Schro...index.html which is listed in the OEIS as A052122. I don't have access to my computer, but it looks like our results for heirarchies of height 1/2 agree. I also have the general solution which checks with OEIS entries for heirarchies of height -2, -1, 1/2, 1, 2, 3 and 4.

The general formula comes from the double binomial expansion,

The formula is reliable, I just computed it for the case s=1/2, to exemplify convergence.

However I just looked in Baker's Paper and indeed he states (as a German native I just translate it):

Quote:Proposition 17. Let ; for each real let be the by

uniquely determined formal series which has the form

.

Then has a positive radius of convergence if and only if is an integer number. is the -th iterate of for integer , hence an entire function. is the inverse series development of for integer .

So instead just of to numerically verify, can we prove that converges for some ?