Down-arrow notation

Down-arrow notation refers to one of the two functions:
# A left-associative extension for addition, multiplication and exponentiation. It is a weaker version of arrow notation, but comparable with arrow notation with the index of the arrow halved.2.03 - The Weak Hyper-Operators. Retrieved 2023-10-11
# The logarithmic inverse of Knuth's up-arrow notation. It is a very slow-growing function and extends the idea of iterated logarithm (log*). https://mathworld.wolfram.com/DownArrowNotation.html
We explain both definitions in this article.
==Formulation 1==
===Definition===
Formally, \(a \downarrow \cdots \downarrow b\) for positive integers \(a\) and \(b\) is defined as follows:
\begin{eqnarray*}
a \downarrow^n b = 
\left\{
\begin{array}{cl}
a^b & (n = 1) \\
a & (n > 1, b = 1) \\
(a \downarrow^n (b-1)) \downarrow^{n-1} a & (n > 1, b > 1)
\end{array}
\right.
\end{eqnarray*}
Here, \(a \downarrow^n b\) for a positive integer \(n\) is a shorthand for \(a \downarrow\cdots\downarrow b\) with n down-arrows.
When \(n = 2\), \(a \downarrow\downarrow b = a^{a^{b-1}}\). The inequality \(a \downarrow\downarrow a = a^{a^{a-1}} < a^{a^a} = a \uparrow\uparrow 3\) is useful when bounding down-arrows in terms of up-arrows.
It can be shown that \(a \downarrow^{2n-1} b \ge a \uparrow^n b\) for \(a, b, n \ge 1\).
Down-arrow notation is not as important in googology as the up-arrow notation, but it is used to distinguish the two possible meanings of fuga- and in the definition of Clarkkkkson. Alistair has assigned fuga- to the down-arrow form, and agreed to rename the larger up-arrow form as megafuga-.
===Examples===
*\(n \downarrow\downarrow n\) = fuga-n, in contrast to \(n \uparrow\uparrow n\) = megafuga-n (Alistair Cockburn)
*\(3 \downarrow\downarrow 3 = ((3 \downarrow\downarrow 1)\downarrow 3)\downarrow 3 = (3^3)^3 = 3^9 = 19,683\)
*\(3 \downarrow\downarrow\downarrow 2 = (3 \downarrow\downarrow\downarrow 1)\downarrow\downarrow 3 = 3\downarrow\downarrow 3\)
*\(3 \downarrow\downarrow\downarrow 3 = (3 \downarrow\downarrow\downarrow 2)\downarrow\downarrow 3 = 19,683 \downarrow\downarrow 3 = 19,683^{19,683^2} = 3^{3^{20}}\)
Using BCalc, we find \(3 \downarrow\downarrow\downarrow 3 \approx 2.198197017816742204069545026183 \cdot 10^{1,663,618,948}\)
===Bounds===
*\(a\downarrow\downarrow\downarrow b\approx a\uparrow\uparrow(b+1)\)
*\(a\downarrow^4 b\approx a\uparrow\uparrow(a(b-1)+1)\)
*\(a\downarrow^5 b\approx(a\uparrow\uparrow)^{b-1}(a(a-1)+1))>a\uparrow\uparrow\uparrow b\)
*\(a\downarrow^6 b\approx(a\uparrow\uparrow)^{(a-1)(b-1)}(a(a-1)+1)>a\uparrow\uparrow\uparrow((a-1)(b-1)+1)\)
==Formulation 2==
===Definition===
Another formulation of down-arrow notation is given as an "inverse" of Knuth's up-arrow notation in some sense. https://mathworld.wolfram.com/DownArrowNotation.html
For example, \(e \downarrow n\) is defined as \(\log n\), and \(e \downarrow \downarrow n\) is defined as the iterated logarithm \(\log^* n\).
==See also==
*fuga-
*Fugagoogolplex
ja:下矢印表記
zh:下箭號表示法
Category:Notations
Category:Functions
Category:Arrow
Category:Primitive recursive