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There are the following common notations:

  • Sums: $$\sum_{i=2}^4 i = 2 + 3 + 4 = 9$$
  • Products: $$\prod_{i=2}^4 i = 2\times 3\times 4 = 24$$

Is there a (theoretical) one for:

  • Exponentiation (conventional/top down) $$\ X_{i=2}^4 i = {2 ^ {3^4}} = 2^{81} =2\,417\,851\,639\,229\,258\,349\,412\,352$$

There is probably no need for of having a special notation for non-conventional Exponents (bottom up) like ${{2 ^ 3}^4} = 2 ^ {3\times4} = 4096$ in the first place. Since this is equivalent to $$n^{\prod_{i=n+1}^4 i} ;n=2$$

Or does this type of notation anyway only make sense for associative and commutative operations; or is this just too exotic?

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    $\begingroup$ This is the closest thing I can come up with. mathworld.wolfram.com/KnuthUp-ArrowNotation.html $\endgroup$ – dylan7 Feb 17 '15 at 15:17
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    $\begingroup$ Probably no notation exists for this because it really has no use. In particular, since exponentiation is neither associative nor commutative, so you can't say $X_{i\in I} x_i$, for example, where $I$ is some index set, unless the index set is well-ordered and finite. $\endgroup$ – Thomas Andrews Feb 17 '15 at 15:24
  • $\begingroup$ Thanks for hinting to Up-Arrow Notation, I think this concept is only somewhat useful in the context of hyperoperation sequences. $\endgroup$ – mxfh Feb 17 '15 at 15:29
  • $\begingroup$ now that I know what a tetration is, this question is probably just the amateur version of Notation for n-ary exponentiation $\endgroup$ – mxfh Feb 17 '15 at 15:38
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    $\begingroup$ After it is obvious that there is no analoguous notation in use, why not simply propose to use big $E$ for this? $E_{i=1}^n i = n^{...^{(i+1)^i}} = n^{...^{2^1}}$ ? $\endgroup$ – Gottfried Helms Feb 21 '15 at 19:12

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