# repeated exponents sign

I'm wondering if there is a exponent version of $\sum$ or $\prod$ or I've even seen a big k used for repeated division. Is there a similar symbol for exponentiation and are there any useful mathematical identities or equations that make use of it. So for instance how could I show $x^{x/2^{x/4^{x/6^\dotsc}}}$

• As far as I know there isn't, and even if there were it wouldn't be particularly useful. Note that exponentiation is not associative, so in order for "exponentiate this list of things" to be well-defined you need to make a convention about how to parenthesize them. – Qiaochu Yuan Aug 17 '15 at 18:16

I've seen this operation represented as

$$\underset{i=1}{\overset{n}{\LARGE\mathrm E}}\;x_i$$

Using this notation, your example would look something like this:

$$\underset{i=1}{\overset{n}{\LARGE\mathrm E}}\;\frac{x}{2i}$$

Note that this is has a different first base than your example ($x$ vs. $\frac{x}{2}$) to simplify typesetting. You can get a little more detail on generic exponentiation on OEIS.

Tetration is a special case that has the form

$$\underset{i=1}{\overset{n}{\LARGE\mathrm E}}\;a$$

You can get more details on tetration on Wikipedia.

• This symbol also assumes that all of the terms repeat. It would be analogous to saying, $n*x$ desscribes all ways to add terms together. But actually, $n*x = \sum_{i=1}^n x$ and all the terms are the same; in general, you want different terms in your sum (the same may be true of the exponentiation). But as Qiaochu pointed out, it's not an associative operation so why bother. – Squirtle Aug 17 '15 at 18:23
• Good point. Edited to indicate that tetration is just a very limited special case of what the OP is asking for. – BunsOfWrath Aug 17 '15 at 18:29
• How can it be difficult to use -for instance- a big E in place for the big $\sum$ resp $\Pi$. I think in wikipedia someone has also proposed a big T (alluding to "T"etration) in place of the $\sum$ resp. $\Pi$ – Gottfried Helms Aug 17 '15 at 18:30