But what if we start not with the Kneser tetration, but with the regular tetration?
We want to continue regular iteration with respect to its base along a path in the upper (resp. lower) half plane. For example along the path \( b=\gamma(t) \), \( t\in (0,1) \) in my previous post.
Now there are two possibilities: Either we can not pass through the Shell-Thron boundary (see a previous discussion here) and we are done anyway, or we have a holomorphic continuation from base \( \sqrt{2} \) to base \( 2\eta-\sqrt{2} \) through the strict upper halfplane.
As the mapping \( M^+(b) \) does not encounter singularities on the path \( \gamma \) we also have a holomorphic continuation in dependence not of the base but of the developing fixpoint of regular tetration, which runs from 2 to the upper primary fixpoint of \( \exp_{2\eta-\sqrt{2}} \). So at the arriving fixpoint regular tetration is not real anymore, and hence different from the Kneser tetration.
(Continuing along the lower half-plane we would arrive at the lower primary fixpoint.)
So in both cases Kneser and regular tetration can not be the same/analytically continued into each other.
Open question remains whether the Kneser tetration can be continued to base \( \sqrt{2} \) yielding real values there.
Yielding a different real-analytic tetration for base \( \sqrt{2} \) based on *both* fixpoints 2 and 4. I think Sheldon mentioned somewhere something where \( \theta \) vanishes towards imaginary infinity, which could give a new tetration.
We want to continue regular iteration with respect to its base along a path in the upper (resp. lower) half plane. For example along the path \( b=\gamma(t) \), \( t\in (0,1) \) in my previous post.
Now there are two possibilities: Either we can not pass through the Shell-Thron boundary (see a previous discussion here) and we are done anyway, or we have a holomorphic continuation from base \( \sqrt{2} \) to base \( 2\eta-\sqrt{2} \) through the strict upper halfplane.
As the mapping \( M^+(b) \) does not encounter singularities on the path \( \gamma \) we also have a holomorphic continuation in dependence not of the base but of the developing fixpoint of regular tetration, which runs from 2 to the upper primary fixpoint of \( \exp_{2\eta-\sqrt{2}} \). So at the arriving fixpoint regular tetration is not real anymore, and hence different from the Kneser tetration.
(Continuing along the lower half-plane we would arrive at the lower primary fixpoint.)
So in both cases Kneser and regular tetration can not be the same/analytically continued into each other.
Open question remains whether the Kneser tetration can be continued to base \( \sqrt{2} \) yielding real values there.
Yielding a different real-analytic tetration for base \( \sqrt{2} \) based on *both* fixpoints 2 and 4. I think Sheldon mentioned somewhere something where \( \theta \) vanishes towards imaginary infinity, which could give a new tetration.