I'm wondering if anyone has a reference or a method for construction a real-analytic function interpolating the exponential factorial, i.e. a function such that $f(x) = x^{f(x-1)}$ and $f(1) = 1$, analogous to the gamma function interpolating the usual factorial.
It's easy to construct a complex-valued solution that should be analytic on $\mathbb{C}\setminus\mathbb{R}$ to the clearly related functional equation $f(z) = z^{(z-1)^{(z-2)^{f(z-3)}}}$ (note that satisfying the exponential factorial equation implies this, but not the other way around). If you look at the power tower $$
f_n(z) =z^{(z-1)^{(z-2)^{(z-3)^{\cdots (z-n)}}}}
$$
the values have period 3 as $n$ approaches infinity, so it seems reasonable that there are 3 limiting analytic functions, each satisfying $f(z)=z^{(z-1)^{(z-2)^{f(z-3)}}}$. (I haven't proven any of this formally, since I was mostly trying to find a real-valued solution and these are not real valued on $\mathbb{R}$) An example computation, here's a plot of the real and imaginary parts of $f_n(1.5)$ for $n$ from 0 to 100 (real part in black, imaginary part in red):
$f_{3n}$ seems to converge pretty rapidly (and similarly for $f_{3n+1}$ and $f_{3n+2}$). But these limiting functions won't be analytic at integer values; in general it will only be $n-1$ times differentiable at $n\in\{1,2,3,\dots\}$. Also, I don't believe they would have the property $f(x)=x^{f(x-1)}$ because of the 3-period nature of the limit.
Does anyone know of a way to construct a real-analytic solution to the equation, or at least a solution that is analytic (but possibly complex-valued) on $\mathbb{R}$?
If it's easier to solve the equation $f(x) = x^{f(x-1)}$ with a different initial value, i.e. $f(1) = c$ for some other value of $c$, I would also be interested to see that.