Half-iteration of x^(n^2) + 1

[Xorter]

I was intrested in the half-iteration of f_n(x) = x^(n^2) + 1 for Large x.

For instance g_n(x) = f_n ^[1/2](x) - x^n.
H_n(x) = f_n ^[1/2](x) / x^n.

For Large x :
Is abs g_n(x) increasing or decreasing with n ?
Is abs H_n(x) decreasing ?

Probably abs g_n is increasing and abs H_n decreasing.

The focus is on integer n and branch structure.

But also if n is real , are these functions analytic in n ?
Perturbation Theory suggests this.

I wonder how these functions look like on the complex plane , especially with resp to n.

Regards

Tommy1729

Okey, I got some Taylor series of the half -iteration of f_n(x) = x^(n^2)+1 by a PARI/gp programme code:
\( (x^{0^2}+1)^{o0.5} = 2 \)
\( (x^{1^2}+1)^{o0.5} = x+0.5 \)
\( (x^{2^2}+1)^{o0.5} = 0.9645+x/2048+5.633x^4+52.85x^8+395x^{12} ... \)
\( (x^{3^2}+1)^{o0.5} = 0.877+x/2048+0.5536x^9+0.03251x^{18}+1.958x^{27} ... \)
I know these are not the best results, but this is that I could get from my programme. Here is the code:

Code:

init()={
default(format,"g0.4");
}

D(z,n)={for(i=0,n-1,z=z');return(z)}

Car(f,dim)={return(subst(matrix(dim,dim,k,j,D(f^(j-1),k-1)/(k-1)!),x,0))}

Decar(M,dim)={
f=0;for(i=1,dim,f+=M[i,2]*x^(i-1));
return(f);
}

Msqrt(B,dim,prec)={
A=matid(dim);
for(i=0,prec,A=(B*A^-1+A)/2);
return(A);
}

I hope it helps you, and you can develope this code. If you can, please share it with me.