The upper superexponential

[Kouznetsov]

Are \( F_{2,3} \) and \( F_{4,3} \) the same two functions in the Bummer post?

Yes. These are the same functions.
Henryk asked me to plot them all versus real argument as a separate post.

Could you comment on how the behavior of the two functions differ in the complex plane?

Yes.
Functions \( F_{4,5} \) and \( F_{4,5} \) entire.
One of them can be obtained from another one, just displacing the argument.

Tetration \( F_{2,1} \) has, as you know, singularities and the cutline; due to the periodicity, there is set of singulatities and cutlines. Function \( F_{2,3} \) can be obtained by translation of tetration \( F_{2,1} \), so, it has similar singulatities.
[/quote]

Do both functions have the same values at z=+/-i*infinity?

No. There is no need to talk about values \( \pm \infty \),
because each of them is periodic and the periods are imaginary.

Do they have the same periodicity?

No. Periods are different:
\( T_2=2\pi \mathrm{i}/\ln(\ln(2))\approx -17.143148179354847104 {\mathrm i}
\)
\( T_4=2\pi \mathrm{i}/\ln(2\ln(2))\approx 19.236149042042854712 {\mathrm i}
\)

Does only one have singularities?

Tetration \( F_{2,1} \) has singularities;
its displacement \( F_{2,3} \) has too.

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 \( F_{2,3}= F_{4,3}(x+\theta(x)) \)?

Yes.

Is the \( \theta(x) \) function analytic?

Yes.
\( \theta(z)=F_{4,3}^{-1}(F_{2,3}(z)) -z \)
is 1-periodic function; it is almost sinusoidal.

Henryk, can we begin to distribute the draft of our paper?
It would answer a lot of questions we provoked with the plot...