Between \(1 < \alpha < \eta\), we can construct tetration entirely from a real valued iterated integral.

The algorithm is written firstly with the identification:

$$

\begin{align}

\alpha \uparrow^0 x &= \alpha \cdot x\\

\alpha \uparrow^n \alpha \uparrow^{n+1} x &= \alpha \uparrow^{n+1}\,x+1\\

\alpha \uparrow^n 1 &= \alpha\\

\alpha \uparrow^{n+1} k &= \alpha \uparrow^n \alpha \uparrow^n \cdots k\,\text{times}\cdots \uparrow^n \alpha

\end{align}

$$

Then there's a Picard type integral sequence to generating this construction:

$$

\begin{align}

\vartheta(w) &= \sum_{k=0}^\infty \left(\alpha \uparrow^{n} k+1 \right)\frac{w^n}{n!}\\

\Gamma(1-x) \left(\alpha \uparrow^n x\right) &= \sum_{j=0}^\infty \left(\alpha \uparrow^n k+1\right) \frac{(-1)^k}{k!(k+1-x)} + \int_1^\infty \vartheta(-w)w^{-x}\,dw \\

\end{align}

$$

Recalling that \(\alpha \uparrow^{n} k+1\) is just \(\alpha \uparrow^{n-1} \cdots \alpha \uparrow^{n-1} \alpha\) with \(k+1\) alphas.

Where this defines a recursive protocol. I have it written here on ArXiv

https://arxiv.org/abs/2106.03935; which details in much more rigor, why this holds.

But this still just produces the standard regular iteration. So it's not a new "real tetration"--it's just a different approach. (Something I think is going to become much more relevant with all the stuff we're working on now with Borel sums!)

The only other real tetration I have to offer, is the beta method. Which only produces smooth values for bases \(b > \eta\), but can produce analytic values within the Shell thron region.

I suggest, my release of the code--beta.gp--if you want to try and implement that in mathematica. It would be hell though. Because I have 48 gbs of ram on my laptop, and I program in GP with that knowledge. Which means a lot of my code is designed for large and robust amounts of value. Though it does work for simple stuff, I imagine it would be tenuous on Wolfram Alpha.... Even in a C shell my code is slow, lol. But if you wait for an hour after writing Sexp(1+z) you'll get the taylor series about \(z=0\) of this function to ridiculous accuracy

And after shamelessly supporting my own work--you should look at Kouznetsov's C++ program. I can locate it down if you want, but I sent you his book, and I believe the link to it is in the book.