Editing Tetration (section)

== Notation ==
There are many different notation styles that can be used to express tetration. Some notations can also be used to describe other [[hyperoperation]]s, while some are limited to tetration and have no immediate extension.

{|class="wikitable"
|+Notation styles for tetration
!scope="col"| Name
!scope="col"| Form
!scope="col"| Description
|-
! scope="row" | [[Knuth's up-arrow notation]]
| <math>\begin{align}
a {\uparrow\uparrow} n \\
a {\uparrow}^2 n
\end{align}</math>
| Allows extension by putting more arrows, or, even more powerfully, an indexed arrow.
|-
! scope="row" | [[Conway chained arrow notation]]
| <math>a \rightarrow n \rightarrow 2</math>
| Allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain.
|-
! scope="row" | [[Ackermann function]]
| <math>{}^{n}2 = \operatorname{A}(4, n - 3) + 3</math>
| Allows the special case <math>a=2</math> to be written in terms of the Ackermann function.
|-
! scope="row" | Iterated exponential notation
| <math>\exp_a^n(1)</math>
| Allows simple extension to iterated exponentials from initial values other than 1.
|-
! scope="row" | Hooshmand notations<ref name="uxp">{{cite journal |author-first=M. H. |author-last=Hooshmand |date=2006 |title=Ultra power and ultra exponential functions |journal=[[Integral Transforms and Special Functions]] |volume=17 |issue=8 |pages=549–558 |doi=10.1080/10652460500422247 |s2cid=120431576}}</ref>
| <math>\begin{align}
  &\operatorname{uxp}_a n \\[2pt]
  &a^{\frac{n}{}}
\end{align}</math>
| Used by M. H. Hooshmand [2006].
|-
! scope="row" | [[Hyperoperation]] notations
| <math>\begin{align}
  &a [4] n \\[2pt]
  &H_4(a, n)
\end{align}</math>
| Allows extension by increasing the number 4; this gives the family of [[hyperoperation]]s.
|-
!scope="row"| Double caret notation
| {{code|a^^n}}
| Since the up-arrow is used identically to the caret (<code>^</code>), tetration may be written as (<code>^^</code>); convenient for [[ASCII]].
|}

One notation above uses iterated exponential notation; this is defined in general as follows:

: <math>\exp_a^n(x) = a^{a^{\cdot^{\cdot^{a^x}}}}</math> with {{mvar|n}} {{mvar|a}}s.

There are not as many notations for iterated exponentials, but here are a few:
{| class="wikitable"
|+Notation styles for iterated exponentials
!scope="col"| Name
!scope="col"| Form
!scope="col"| Description
|-
!scope="row"| Standard notation
| <math>\exp_a^n(x)</math>
| [[Leonhard Euler|Euler]] coined the notation <math>\exp_a(x) = a^x</math>, and iteration notation <math>f^n(x)</math> has been around about as long.
|-
!scope="row"| Knuth's up-arrow notation
| <math>(a{\uparrow}^2(x))</math>
| Allows for super-powers and super-exponential function by increasing the number of arrows; used in the article on [[large numbers]].
|-
!scope="row"| Text notation
| {{code|2=tex|exp_a^n(x)}}
| Based on standard notation; convenient for [[ASCII]].
|-
!scope="row"| J Notation
| {{code|2=j|x^^:(n-1)x}}
| Repeats the exponentiation. See [[J (programming language)]]<ref>{{cite web |title=Power Verb |url=http://www.jsoftware.com/help/dictionary/d202n.htm |work=J Vocabulary |publisher=J Software |access-date=28 October 2011}}</ref>
|-
!scope="row"| Infinity barrier notation
| <math>a\uparrow\uparrow n|x</math>
| Jonathan Bowers coined this,<ref>{{cite web |title=Spaces |url=http://www.polytope.net/hedrondude/spaces.htm |access-date=17 February 2022}}</ref> and it can be extended to higher hyper-operations
|}