For considerations of the characteristic of the powerseries of U-tetration I recently uploaded an empirical table, but this link may have been overlooked in our threads. Here it is again:
coefficients for fractional iteration
I think, I've to correct my estimation there about growth of absolute values of terms. Better estimation (instead of hypergeometric) seems to be |term_k| ~ const*exp(k^2) asymptotically by inspection of first 96 terms with fractional heights. There is a "bump" at one index k, from where the absolute values grow after they have initially decreased. This "bump" moves to higher k with |1/2-fractional(h)|-> 1/2, and maybe we can say, it moves out to infinity, if h is integer, and the beginning of growth of absolute values of terms does not occur anywhere.
An older and shorter treatize of height-dependent coefficients is in
coefficients depending on h (older) Unfortunately I choose the letter U for the matrix, which would be the matrix "POLY" in my article, so this should no more be confusing (I'll change this today or tomorrow)
Gottfried
coefficients for fractional iteration
I think, I've to correct my estimation there about growth of absolute values of terms. Better estimation (instead of hypergeometric) seems to be |term_k| ~ const*exp(k^2) asymptotically by inspection of first 96 terms with fractional heights. There is a "bump" at one index k, from where the absolute values grow after they have initially decreased. This "bump" moves to higher k with |1/2-fractional(h)|-> 1/2, and maybe we can say, it moves out to infinity, if h is integer, and the beginning of growth of absolute values of terms does not occur anywhere.
An older and shorter treatize of height-dependent coefficients is in
coefficients depending on h (older) Unfortunately I choose the letter U for the matrix, which would be the matrix "POLY" in my article, so this should no more be confusing (I'll change this today or tomorrow)
Gottfried
Gottfried Helms, Kassel