Negative, Fractional, and Complex Hyperoperations

[MphLee]

\( s \)-rank hyperoperations have meaning as long as we can iterate \( s \) times a function \( \Sigma \) defined in the set of the binary functions over the naturals numbers (or defined over a set of binary functions.)

let me explain why.

There are many differente Hyperoperations sequences, end they are all defined in a different way:

we start with an operation \( * \) and we obtain its successor operation \( *' \) applying a procedure \( \Sigma \) (usually a recursive one).

\( \Sigma(*)=*' \)

So every Hyperoperation sequence is obtained applying that recursive procedure \( \Sigma \) to a base operation \( * \) (aka the first step of the sequence)

\( *_0:=* \)
\( *_1:=\Sigma(*_0) \)
\( *_2:=\Sigma(\Sigma(*_0)) \) and so on

or in a formal way

\( *_0:=* \)
\( *_{n+1}:=\Sigma(*_n) \)

That is the same as

\( *_{n}:=\Sigma^{\circ n}(*_{0}) \)

so if we can extend the iteration of \( \Sigma^{\circ n} \) from \( n \in \mathbb{N} \) to the real-complex numbers the work is done.

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