Tetration series for integer exponent. Can you find the pattern?

[marraco]

So your basic problem is the half iterate isn't analytic at b=1, no matter which approach you use, Kneser or the simpler Schroeder function solution. For simplicity, I'm graphing the Schroeder function half iterate as you loop around from 1.1 to -0.9 and then back to 1.1, but the Kneser solution you get using both fixed points (which is what fatou.gp gives you) isn't visually different for this graph. If it isn't analytic, increasing the precision of the number of sample points or decreasing the radius doesn't help.

I do not understand if you speak of "half iterate", because your code uses it internally, or you say that I'm looking for the Taylor series of \( \\[15pt]

{^{0.5}x} \) (I'm looking for \( \\[15pt]

{^{1.5}x} \)).

Now, 1.5 has been choosen by me arbitrarily. The Taylor series exist around 1 for integer values (the thread starts with the explicit formulas of those Taylor series).
I could had chosen any exponent =m+1/n (m and n integer).

Of course (I understand), knesser (and fatou?) give the Taylor series for a constant base \( \\[15pt]

{^{x}b, \;\; b \neq 1} \), and I don't need the Taylor series of \( \\[15pt]

{^{x}b} \) around 1. I already know that \( \\[15pt]

{^{1.5}1=1} \), and I need high precission calculating \( \\[15pt]

{^{1.5}(1+n.\delta)} \)

Because \( \\[15pt]

{(1+n.\delta)} \) is not 1, then the Taylor series should exist there, no matter if there is a singularity at 1.

Do you mean that is not possible to get higher precision for \( \\[15pt]

{^{x}(1+n.\delta)} \), with (nonlinear) conjugate gradient using knesser as initial guess?