# Concise notation for iterated exponentiation involving an unknown

I am working with some tetration problems, such as below: $$y = e^{e^x}$$

and I am looking for a concise notation for this. In particular, I would like a way to indicate $$n$$ iterations of the exponentiation, with the deepest level raised to $$x$$, rather than $$e$$.

My initial thoughts were to write the above example as $$e^{e^x} = ({^2e})^x = (e\uparrow\uparrow2)^x$$. However, since exponential towers must be evaluated from top to bottom, it seems like this is not true.

Is there any other concise notation for this?

Thanks.

• Indeed, have a look into Knuth's superpower notation. – Wuestenfux Sep 21 '18 at 12:23
• What about "exp(exp(x))" ? – Peter Sep 21 '18 at 12:24
• The $n$-th iterate of $f$ is usually denoted $f^{(n)}$, so I guess $\mathrm{exp}^{(n)}(x)$ would do. – Michal Adamaszek Sep 21 '18 at 13:17

Suggestion 1: In the tetration forum we have partly used $$\exp^{\circ h}_b(x)$$ (the little circle indicating function-composition instead of powers or instead of derivation) for the general iteration $$x \to b^x$$ to the iteration-(h)eight $$h$$ of the exponentialtower. I myself use sometimes $$\text{T}^{\circ h}_b(x)$$ for shortness and $$\text{U}^{\circ h}_b(x)$$ for the decremented exponentiation $$x \to b^x−1$$.

Suggestion 2: In many articles I've also seen the simple solution to use the index-notation. So $$z_0$$ for the initial value , $$z_1=b^{z_0}$$ then $$z_h=b^{z_{h−1}}$$ for the $$h$$'th iteration (exponentialtower of (h)eight $$h$$) and $$z_\infty$$ if that limit exists. In articles the base $$b$$ is mostly a fixed parameter over a lot of formulae and algebraic derivations so I'd prefer such a notation which allows to omit this reference to $$b$$ to reduce redundancy in notation. (I find this unbeatable concise - unfortunately the indexing-notation indicates many things in math so I use this only when I'm well sure it is not obfuscating my line of discussion/derivation/definition)