About Hyper-operators

Hyper-operators or hyper-operations, are members of the hyper-operation sequence, of which the first few members are: addition, multiplication, exponentiation, tetration, pentation, hexation, and so on. The first three are very well understood, but the later operations are not so well understood, but since they are defined in terms of the previous operation, at least we know integer values.

The hyper-operation sequence has also been known as the Ackermann function and the Grzegorczyk hierarchy. The most notable sources that use this terminology are Daniel Geisler's Tetration.org and Rubstov and Romerio's paper: Ackermann's function and New Arithmetical Operations respectively. Since "hyper-operations" is the generic name for this sequence, it is the terminology that is used here. Rubstov and Romerio's paper is also where I got the terms super-logarithm and super-root.

Values of Hyper-operators

The hyper-operations sequence is denoted by x⁽ⁿ⁾y, and although there is ongoing research into extending it to all real arguments, the domain of the ternary function associated with it is Z³ (all integers). Even with this definition, there are quite a few properties that can be gleaned. Most of the time, only n is considered as an integer, because extensions to complex (which includes real) x, y have been defined for n = 1, n = 2, and n = 3. Here are some values at fixed n:

`x`⁽⁰⁾`y` = `x` ° `y`	"zeration" (the successor function in most cases, see Rubstov's paper at the top)
`x`⁽¹⁾`y` = `x` + `y`	"addition"
`x`⁽²⁾`y` = `x` `y`	"multiplication"
`x`⁽³⁾`y` = `x`^y = ¹`x`^y = e^{y log(x)}	"exponentiation"
`x`⁽⁴⁾`y` = ^y`x` = ^y`x`¹	"tetration"

There are also more interesting properties at fixed values of x and y. Some of these properties could possibly be used to extend hyper-operations to real n:

`x`⁽ⁿ⁾0 = 1	for `n` ≥ 3
`x`⁽ⁿ⁾1 = `x`	for `n` ≥ 2
2⁽ⁿ⁾2 = 4	for `n` ≥ 1
`x`⁽ⁿ⁾`x` = `x`⁽ⁿ⁺¹⁾2	for `n` ≥ 1

Hyper-operator Derivatives

Since hyper-operators are all binary operators (2 arguments), I will be using the gradient (written "∇") for the derivatives instead of partial derivatives with respect to each argument. The gradient is just a vector that denotes the partial derivatives with respect to all the arguments of a scalar function.

∇ f (x, y) = [ ∂/∂x, ∂/∂y ] f (x, y)

the gradient of a binary function

These are the gradients of the first few hyper-operators:

∇ (`x`⁽¹⁾`y`) = ∇ (`x` + `y`) = [ 1, 1 ]	the gradient of addition
∇ (`x`⁽²⁾`y`) = ∇ `xy` = [ `y`, `x` ]	the gradient of multiplication
∇ (`x`⁽³⁾`y`) = ∇ `x`^y = [ `y` `x`^y-1, `x`^y log(`x`) ]	the gradient of exponentiation
∇ (`x`⁽⁴⁾`y`) = ∇ ^y`x` = [ ^y`x` (^y-1`x`/`x` + log(`x`) ∂/∂`x` (^y-1`x`)), ^y`x` (log(`x`) ∂/∂`y` (^y-1`x`)) ]	the gradient of tetration

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