A nice series for b^^h , base sqrt(2), by diagonalization

[Gottfried]

Don't know yet, whether this has some benefit so far.

It looks, as if we had a discussion of that recently in Upper superexponential

I'm excerpting a bit of Henryk's post:

As it is well-known we have for \( b<e^{1/e} \) the regular superexponential at the lower fixed point.

This can be obtained by computing the Schroeder function at the fixed point \( a \) of \( F(x)=b^x \).

(...)
Now the upper regular superexponential \( \operatorname{usexp} \) is the one obtained at the upper fixed point of \( b^x \).
For this function we have however always \( \operatorname{usexp}(x)>a \), so the condition \( \operatorname{usexp}(0)=1 \) can not be met.
Instead we normalize it by \( \operatorname{usexp}(0)=a+1 \), which gives the formula:
(*1) \( \hspace{48} \operatorname{usexp}_b(t)=a+\chi^{-1}\left(\ln(a)^x \chi(1)\right) \)
(...)

My construction in the previous post was obviously the same as that above construction (*1) ... Gottfried (I added the comments //... )

Then we can write
\(
\hspace{48} C = \sum_{j=0}^{\infty} c_j*(-1/2)^j \hspace{96} \text{//this is Schroeder-function\ }\chi_2(x) \text{\ for \ } 2^x - 1 \text{\ at \ } x=-\frac12} \\
\hspace{48} \exp^{\circ h}_{\sqrt{2}}(1) =2 + \sum_{k=1}^{\infty} 2* C^k * d_k * v^k \hspace{24} // = 2+2*\chi_2^{-1}(u^h*\chi_2(-1/2)) \\
\)
and the k'th coefficient in my first mail is just 2* C^k * d_k in the formula above.

where the fixpoint "a" is simply given as constant 2 and could be generalized to the symbol. The sum-expression describes the inverse of the schröder-function chi^-1 in Henryk's post. The formula for the repelling fixpoint replaces simply 2 by 4 and (1/2-1) by (5/4-1) and uses the adapted schröder-function. So I think it's useful to redirect replies to the other thread...