The upper superexponential

[sheldonison]

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:
\( \operatorname{usexp}_b(t)=a+\chi^{-1}\left(\ln(a)^x \chi(1)\right) \)

The "upper/lower" properties of these two sexp solutions are very interesting, especially being able to convert one to the other. The "upper" solution approaches the larger fixed point at -infinity, and the lower solution approaches the smaller fixed point at +infinity.

Can this be applied to Kneser's fixed point solution for bases larger than (e^(1/e))? For base e, Kneser's solution, has complex values at the real number line, and the function approaches the fixed point as x grows towards +infinity. But the desired solution has real values for all x>-2, and complex values for all x<-2 (except for the singularities). Moreover, the desired solution approaches the fixed point, as real x approaches -infinity.

This has probably already been done, but can Kneser's base e solution, approaching a complex fixed point at +infinity, be converted it to another solution, approaching the fixed point at -infinity, with real values at the real number line, for all x>-2? Perhaps this line of reasoning isn't applicable because the resulting solution, approaching the fixed point at -infinity, probably would not have imaginary values of zero for for real all x>-2.
- Sheldon