Editing Knuth's up-arrow notation (section)

==Definition==

Without reference to [[hyperoperation]] the up-arrow operators can be formally defined by

:<math>
  a\uparrow^n b=
   \begin{cases}
    a^b, & \text{if }n=1; \\
    1, & \text{if }n>1\text{ and }b=0; \\
    a\uparrow^{n-1}(a\uparrow^{n}(b-1)), & \text{otherwise }
   \end{cases}
 </math>
for all integers <math>a, b, n</math> with <math>a \ge 0, n \ge 1, b \ge 0</math>.<ref group="nb" name=corona2/>

This definition uses [[exponentiation]] <math>(a\uparrow^1 b = a\uparrow b = a^b)</math> as the base case, and [[tetration]] <math>(a\uparrow^2 b = a\uparrow\uparrow b)</math> as repeated exponentiation.  This is equivalent to the [[Hyperoperation#Definition|hyperoperation sequence]] except it omits the three more basic operations of [[Successor_function|succession]], [[addition]] and [[multiplication]].

One can alternatively choose [[multiplication]] <math>(a\uparrow^0 b = a \times b)</math> as the base case and iterate from there. Then [[exponentiation]] becomes repeated multiplication. The formal definition would be
:<math>
  a\uparrow^n b=
   \begin{cases}
    a\times b, & \text{if }n=0; \\
    1, & \text{if }n>0\text{ and }b=0; \\
    a\uparrow^{n-1}(a\uparrow^{n}(b-1)), & \text{otherwise }
   \end{cases}
 </math>
for all integers <math>a, b, n</math> with <math>a \ge 0, n \ge 0, b \ge 0</math>.

Note, however, that Knuth did not define the "nil-arrow" (<math>\uparrow^0</math>). One could extend the notation to negative indices (n ≥ -2) in such a way as to agree with the entire hyperoperation sequence, except for the lag in the indexing:
:<math>H_n(a, b) = a [n] b = a \uparrow^{n-2}b\text{ for } n \ge 0.</math>

The up-arrow operation is a [[Associative property#Notation for non-associative operations|right-associative operation]], that is, <math>a \uparrow b \uparrow c</math> is understood to be <math>a \uparrow (b \uparrow c)</math>, instead of <math>(a \uparrow b) \uparrow c</math>. If ambiguity is not an issue parentheses are sometimes dropped.