Is there official notation to represent "perform an operation n times"?

Question

I would like to know if you can represent the idea of say,

$$n^n$$

n amount of times without defining a function. For example:

$$1$$ $$2^2$$ $$3^{3^3}$$ $$4^{4^{4^4}}$$ $$5^{5^{5^{5^5}}}$$

and so on, n amount of times. Is there a way to represent this mathematically without defining a function?

EDIT: I understand that in this particular example the answer is tetration. However, I am searching for more general notation that could also work with the following:

$$\ln(n)$$ $$n!$$ $${^n}n$$

all n amount of times. Another example with factorial:

$$1!$$ $$(2!)!$$ $$((3!)!)!$$ $$(((4!)!)!)!$$ $$((((5!)!)!)!)!$$

Answer 1

It sounds like you are looking for the notation for an iterated function. If $f$ is the function you want to iterate, you generally write $$f^n$$ for the $n^\text{th}$ iterate of $f$. (Notice that this is different from the notation for the $n^\text{th}$ derivative of $f$, which is given by $f^{(n)}$). So, for example, you could write $$f(n) = n!$$ and then $$f^1(n) = n!$$ $$f^2(n) = (n!)!$$ and so on.

Answer 2

You can use Knuth's up-arrow notation:

$a\uparrow n=a^n$

$a\uparrow \uparrow n=a^{a^{a^{\dots}}} $ n times

$a\uparrow \uparrow \uparrow n= a\uparrow \uparrow a\uparrow \uparrow a\uparrow \uparrow \dots$ n times

and in general

$a\uparrow ^b n=a\uparrow ^{b-1} a$ iterated n times

Answer 3

To add to Strants' answer, which indeed gives the most common way of writing "do $f$ to an argument $n$ times", I'll give the "expanded form" of this which is $$f^n(x) \equiv f(f(\stackrel{(n)}{\cdots}f(x))$$

This is cumbersome and I strongly recommend using the compact notation, even if it means defining a new function. However it might suit your purposes for cases where the function has a reserved definition for the superscript (e.g. $\sin^2$), or you don't want to introduce a new function name. So for example, $$\sin(\sin(\stackrel{(n)}{\cdots}\sin(x)))$$

would apply the sine function $n$ times to $x$. However, take for example $$f(x) = \frac{2x+1}{x^2+e^x}$$

The $n$-iterated expression for it is (for me) unimaginable, and even if presented with it I would most likely not know at all what it was meant to designate. Of course, I could always write $f^n(x)$ and not have a doubt in the world :)