# Tetration Forum

Full Version: Bell formula for iterated exponentiation
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Gottfried Wrote:m=arbitrary, integer>0:
$\hspace{24} T^{\tiny(m)}_s(x)=
\sum_{ k_m...k_1 =0..\infty \\ k_1+k_2+...+k_m=n }
x^{k_m}
* \left( k_{m-1}^{k_m} \dots k_2^{k_3} k_1^{k_2} \right)
* \begin{pmatrix} n \\ k_2,k_3,...,k_m \end{pmatrix}
* \frac{ log(s)^n}{n!}
$

(hope I didn't make an index-error).
This formula can also be found in
E.T.BELL ,"The iterated Exponential Integers", Annals of Mathematics (193 p539-557
Are you sure Gottfried? I didnt find it in the mentioned article.
bo198214 Wrote:Are you sure Gottfried? I didnt find it in the mentioned article.

Too bad. I don't know, where my brain was... I had just skimmed through some articles and meant to have caught the source correctly. I'll seek, where I've actually have seen it, maybe in Abramowitsch/Stegun...
Anyway, it is easy to derive it with pen & paper.

I'll be back in the evening, going for a walk today -

Gottfried