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So far, we use the symbol $$\sum$$ to denote sums, and $$\prod$$ to denote products. But is there any such notation for exponentiation?

Has any research been done about exponentiation of this type, where the numbers in the "power tower" form some sequence? $$a_1^{{a_2}^{{...}^{a_n}}}$$ And it need not be a finite sequence, I might add...

Does anybody know of any examples that we know how to evaluate? Does anybody know how to determine convergence or divergence in these "power towers"? And can anybody suggest or point me towards any existing way to denote something like this?

Let me give an example. Suppose we have the power tower defined by the sequence $$a_n=\frac{3}{2^n}$$ Does anybody know how to evaluate the infinite power tower defined by this sequence (it seems to converge) ? How about a partial one?

We already know how to evaluate some infinite power towers where $a_n$ is a constant, like $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdots}}}=2$$

Does anybody know of any resources about this to point me towards, or any original insights or ideas about this concept?

Thanks!

Franklin Pezzuti Dyer
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3 Answers3

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Creutz/Sternheimer ([5] and [6] below) use $\Xi$ for the exponentiation analog of $\Sigma$ (summation) and $\Pi$ (product). They were probably unaware of [4], but a similar usage appears in [4] for the corresponding transfinite operations on ordinals --- $\Xi_{0}$ is for $\Sigma$ (transfinite sums), and $\Xi_{1}$ is for $\Pi$ (transfinite products), and $\Xi_{2}$ is for transfinite "bottom-up evaluated" exponential towers of ordinals, and $\Xi_{3}$ is for transfinite "bottom-up evaluated" bottom-up-tetration towers of ordinals, etc. (When defining these higher operations for ordinals, the top-down evaluation method does not work very well -- see this question and my comments under it.)

The papers below (except for [4]) are the most relevant papers I know of for what you are mainly asking about, namely convergence issues for limits of power towers of the form $a_1^{{a_2^{{...}^{{a_n}^{...}}}}}.$

[1] David Francis Barrow (1888-1970), Infinite exponentials, American Mathematical Monthly 43 #3 (March 1936), 150-160.

[2] Wolfgang Joseph Thron (1918-2001), Convergence of infinite exponentials with complex elements, Proceedings of the American Mathematical Society 8 #6 (December 1957), 1040-1043.

[3] Donald Lewis Shell (1924-2015), On the convergence of infinite exponentials, Proceedings of the American Mathematical Society 13 #5 (October 1962), 678-681.

[4] Arthur Leonard Rubin (1956-$_{---}$) and Jean Estelle Rubin (1926-2002), Accumulation functions on the ordinals, Fundamenta Mathematicae 70 #2 (1971), 205-220.

[5] Michael John Creutz (1944-$_{---}$) and Rudolph Max Sternheimer (1926-2000), On the convergence of iterated exponentiation---I, Fibonacci Quarterly 18 #4 (1980), 341-347.

[6] Michael John Creutz (1944-$_{---}$) and Rudolph Max Sternheimer (1926-2000), On the convergence of iterated exponentiation---II, Fibonacci Quarterly 19 #4 (1981), 326-335.

[7] Irvine Noel Baker (1932-2001) and Philip Jonathan Rippon (??-$_{---}$), Iterating exponential functions with cyclic exponents, Mathematical Proceedings of the Cambridge Philosophical Society 105 #2 (March 1989), 357-375.

Dave L. Renfro
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Let, $$h(z)=z^{z^{z^{\cdots}}}.$$ Now we have, $$h(z)=\frac{W(-\ln z)}{-\ln z}$$ Where $W(y)$ is the Lambert W function. Maybe this can give you some ideas, who knows :)

Note : If we plug $z=\sqrt{2}$ in the formula we get the result $h(z)=2$.

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here's my research (somewhat as it goes), if you have a finite power tower like:

$a^{{(ab)}^{(ab^2)}}$

then we have $(ab)^{(ab^2)}$ as a power by the power rules used for integers at least this equals:

$$a^{(ab^2)}b^{(ab^2)}=(a^a)^{(b^2)}(b^{(b^2)})^a=({(a^a)^b})^b({(b^b)^b})^a$$ this is our exponent if we then raise a to it we get: $$\Large a^{(a^{(ab^2)}b^{(ab^2)})}=a^{((a^a)^{(b^2)}(b^{(b^2)})^a)}=a^{(({(a^a)^b})^b({(b^b)^b})^a)}= (a^{({(a^a)^b})^b }\Large )^{({(b^b)^b})^a}= ???$$

it's a work in progress I guess for me. this example was for a geometric sequence.