Definition:Hyperoperation/Nth Hyperoperation

Definition

Let $\left\langle{H_n}\right\rangle$ denote the hyperoperation sequence.

The $n$th term of $\left\langle{H_n}\right\rangle$, which is the binary operation $H_n : \Z_{\ge 0} \times \Z_{\ge 0} \to \Z_{\ge 0}$, is known as the $n$th hyperoperation.

$\forall x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases} y + 1 & : n = 0 \\ x & : n = 1, y = 0 \\ 0 & : n = 2, y = 0 \\ 1 & : n > 2, y = 0 \\ H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \end{cases}$

Also known as

Some sources refer to a hyperoperation as a hyperoperator function.

Also see

• Definition:Ackermann Function
• Definition:Ackermann-Péter Function