# Attempt on fractional tetration

To start off, I was looking at the following ingeniously made form of the Gamma function:

$$\Gamma(z+1)=\lim_{n\to\infty}\frac{n!(n+1)^z}{(1+z)(2+z)\cdots(n+z)}$$

which lies on the back of

$$1=\lim_{n\to\infty}\frac{n!(n+1)^z}{(n+z)!}$$

for all integer $z$. One them multiplies through by $z!$ and use the recursive formula for the factorial to reach the above formula.

In the same light, I was wondering if a limit definition of tetration was possible. Consider the following:

$$a^{a^{a^{\dots}}}=\underbrace{a\uparrow a\uparrow a\uparrow \dots\uparrow}_nb=a\uparrow_nb$$

And then consider the following:

$$a\uparrow_nf(n)$$

Particularly, I was wondering if there was a continuous function $f:\mathbb R\to\mathbb C$ such that

$$\lim_{n\to\infty}a\uparrow_nf(n)=c$$

for some constant $c$. From there, one could imagine something like...

\begin{align}a\uparrow_{1/2}c&=a\uparrow_{1/2}\lim_{n\to\infty}a\uparrow_nf(n)\\&=\lim_{n\to\infty}a\uparrow_{n+1/2}f(n)\\&=\lim_{n\to\infty}a\uparrow_nf(n-\frac12)\end{align}

Does this seem reasonable? Does anyone know much about if this is a good path towards defining fractional ordered tetration?