Iterability of exp(x)-1

[bo198214]

Now I indeed had a look at

[1] P. L. Walker, A class of functional equations which have entire solutions, Bull. Austral. Math. Soc. 38 (198, no. 3, 351-356

but things become more complicated!
Lets starting with his theorem:

Theorem 2. Let \( \phi \) be an entire function of the form \( \phi(z)=z+\sum_{n=1}^\infty c_n z^{n+1} \), where \( c_1>0, c_n\ge 0 \) for all \( n \), and either (i) \( c_2\neq c_1^2 \) or (ii) \( c_3<c_1^3 \).
Then the sequence \( (f_n) \) defined in Theorem 1 converges uniformly on every \( \overline{S}(0,M) \) to a function \( f \) which is an entire non-constant solution of (2).

Where (2) is \( f(w+1)=\phi(f(w)) \). The also mentioned theorem 1 and sequence \( f_n \) does not matter yet.

What however really bothers me, that it seems not to be true:
Let \( \phi(x)=x+x^2 \). This is a feasible function for theorem 2, with \( c_1=1 \) and \( c_2=0\neq c_1^2 \).
Now I looked at the (unqiue) half iterate
\( \phi^{\circ 1/2}(x)=f(f^{-1}(x)+1/2) \)
which should be entire too, for comparison some members of its series:
\(
x+{\frac {1}{2}}{x}^{2}-{\frac {1}{4}}{x}^{3}+{\frac {1}{4}}{x}^{4}-{
\frac {5}{16}}{x}^{5}+{\frac {27}{64}}{x}^{6}-{\frac {9}{16}}{x}^{7}+{
\frac {171}{256}}{x}^{8}-{\frac {69}{128}}{x}^{9}+O \left( {x}^{10}
\right)
\)
and tested convergence at \( x=5 \) (an entire function has an infinite radius of convergence, so it should converge for every \( x \)) and what did I find? Divergence!

0, 5.0, 17.50000000, -13.75000000, 142.5000000, -834.0625000, 5757.734375, -38187.57812, 222737.7149, -830118.7301, -3591005.876, 123831803.7, -1672085945.

So it seems this proof is also not reliable...
Boy that shakes my trust in professional mathematics.