# Fractional exponentiation with different bases

The standard analytic tetration base $b>e^{1/e}$, $F_b$ is the unique analytic function defined everywhere on $ℂ$ except on the ray from -2 to -∞ such that $F_b(0)=1$, $F_b(\overline z) = \overline{F_b(z)}$ and $F_b(z+1)=b^{F(z)}$, and $\lim_{y→∞} F_b(x+iy) = L_b$ where $L_b$ is the unique fix-point of base $b$ logarithm in the upper half-plane (Paulsen and Cowgill 2017).

The $c$th iteration of exponentiation base $b$, $\mathrm{exp}_b^c(x) = F_b(F_b^{-1}(x)+c)$.

Let $a$ and $b$ be real numbers above $e^{1/e}$, and $c < d$ be real numbers. Do we have $\lim_{x→∞}\frac{\mathrm{exp}_a^c(x)} {\mathrm{exp}_b^d(x)} = 0$ ?
(Note that either the limit is 0 or it does not exist.)

In other words, we want to know whether if $c < d$, then $c$th iteration of exponentiation grows asymptotically slower then $d$th iteration of exponentiation, even for fractional $c$ and $d$ and exponentiation using different bases (that are $>e^{1/e}$).

A broader question about naturalness of growth rates related to tetration is linked here. The relation with this question is that if the above limit holds, it gives evidence that fractional exponentiation provides a natural growth rate intermediate between (essentially) quasipolynomial and quasiexponential. Conversely, if the limit does not hold, then unless we have a reason to prefer a particular base (or use another construction altogether), that suggests that despite existence of analytic tetration, we have not convincingly found a natural approximately half-exponential growth rate (if there is one).

• I contacted an author of the cited paper (William Paulsen), and he conjectures the answer to be yes. – Dmytro Taranovsky Nov 22 '17 at 2:43
• However, since we do not have a proof, the question remains open. – Dmytro Taranovsky Nov 22 '17 at 2:44
• The answer to this question turns out to be far more interesting than one would expect. If one assumes Kneser's solution for real values of c and d, then there are cases where the two bases a<>b, and for arbitrarily large x then even if c<d, sometimes $$\exp_a^c(x)>\exp_b^d(x)$$ This can only happen if c is only a little bit smaller than d, say 0.0001 or so. I will post more details later, and give some counter examples. This is somewhat related to a mathstack question I posted some years ago; math.stackexchange.com/questions/208996/half-iterate-of-x2c – Sheldon Mar 27 '18 at 10:39