I would like to know what is known about Taylor series for tetration (and other hyper-exponentiations).
Surprisingly, such information is rare on internet. Numerical values for expansion in hyperexponent for tetration with basis $e$ can be found here:
http://en.citizendium.org/wiki/Tetration#Taylor_expansion_at_zero
However I am interested in expansion in the basis of tetration. I look for
$$ x \uparrow \uparrow m = \sum_{n=0}^{\infty} c_n(m)x^n $$
Wolfram mathematica gives me wired result (obviously a bug) at $x=0$ for $x \uparrow 3$:
https://www.wolframalpha.com/input/?i=taylor+series+x%5E(x%5Ex)
where in the output $log(x)$ appears (not a polynomial).
It is better at $x=1$
https://www.wolframalpha.com/input/?i=taylor+series+x%5E(x%5Ex)+at+x%3D1
I imagine the point $x=0$ may be "peculiar", but at least expansion at $x=1$ should be possible.
List of my questions:
1) Is tetration analytic at $x=0$? At $x=1$?
2) If yes, is an explicit closed-form formula for $c_n(m)$ known? (at any of these points)
3) If not, is an explicit closed-form formula for $c_n(m)$ know for some specific values of $m$?
4) Same questions for extension to higher hyper-exponentiations...
Thank you.
Consider the tetration of the function $e^x$ $$^n(e^x)=(e^x)_1^{(e^x)_2^{(e^x)_3^{_..^{.e^x_n}}}}$$
For a natural number n, the taylor series of that function is
$$^n(e^x)=\sum_{k}^{\infty}\frac{1}{k!}*T(n,k)*x^k$$
where $T(n,k)$ is the OEIS A210725 (if k<n, $T(n,k)=T(k,k)$)
then, for $n\in\mathbb{N}$:
$$^n(x)=\sum_{k}^{\infty}\frac{1}{k!}*T(n,k)*(ln(x))^k$$
On the limit for $n\to\infty$, the T(n,k)=T(k,k) are the coefficients of the Lambert w function.
There are other series, that you can find here.
For code able to calculate tetrations with high precision, (for a limited set of bases, and $n\in\mathbb{R}$) check Sheldonison's fatou.gp code
In Wolframalpha, you can get the series of $^n(x+1)$, and it will produce the Stirling transform of $^n(e^x)$. The series has power of x, and no logarithms.