The upper superexponential

[sheldonison]

... we present here the super-exponentials \( F_{2,3} \) and \( F_{4,3} \). Which however are not distinguishable at the real axis:
...
We tried to make the range of holomorphism of these fuinctions so large as possible, in order to exclude the functions that can be obtained from function \( F \) with modification of the argument: \( \mathcal{F}(x)=F(x+\theta(x)) \), where \( \theta \) is 1-periodic function, holomorphic at least in some vicinity of the real axis, such that \( \theta(0)=0 \). The mofigied function \( \mathcal{F} \) satisfies the same equation; \( \mathcal{F}(x+1)=\sqrt{2}^{\mathcal{F}(x)} \). However, while \( \theta \) is not identically zero, the modification destroys the periodicity of function, allthough the function \( \mathcal{F} \) may be also analytic and even entire (for the case \( L=4 \)). Therefore, we need to keep the periodicity as the criterion, necessary for the uniqueness of these functions. As the periods are imaginary, the extension of functions to the complex plane seems to be the only way to provide the uniqueness.

Functions \( F \) above are related: \( F_{2,3} \) can be expressed through \( F_{2,1} \) and \( F_{4,3} \) can be expressed through \( F_{4,5} \) with some complex constant ofsets of the arguments.

First of all, thanks for the post, this is very interesting material. Are \( F_{2,3} \) and \( F_{4,3} \) the same two functions in the Bummer post?

Could you comment on how the behavior of the two functions differ in the complex plane? Do both functions have the same values at z=+/-i*infinity? Do they have the same periodicity? Does only one have singularities?

Given that \( F_{2,3}(z)= F_{4,3}(z) \) at all integer values of z, then can these two functions be expressed in terms of each other, where \( F_{2,3}= F_{4,3}(x+\theta(x)) \)? Is the \( \theta(x) \) function analytic?

- Sheldon