Code:
- Approaches to tetration
We may classify the approaches to tetration in two classes: a "binary operator
approach" and an "iterative series approach"
-- the operator approach ----------------------------------
Here one tries to extend the hierarchy of binary operator
a+b : addition a*b : multiplication a^b : exponentiation a^^b : tetration
in a meaningful way
Extensions of this are made in two ways
assigning an index to the operation and to evolve this
to higher or lesser indices
a[0]b : "zeration" a[1]b : addition a[2]b : multiplication
a[3]b : exponentiation a[4]b : tetration a[5]b : pentation
...
[see links]
* Ackermann-function
Closely related is the concept of the Ackermann-function
A(a,index,iterate)
[see: Ackermann-function, <literature>]
* Reihenalgebra
Even an approach to extend this operator-hierarchy to fractional indices is known
where the index was adapted:
a[1]x : "do nothing"
a[2]x : add one to a a[1/2]b : subtract one // unary operator
a[3]b : addition a[1/3]b : subtraction
a[4]b : multiplication a[1/4]b : division
a[5]b : exponentiation a[1/5,subscript]b : inverses root or logarithm
a[6]b : tetration a[1/6,subscript]b : inverses depending on evaluation-precedence
...
The author tried then to find meaningful interpolations for this indexing scheme.
[markus Müller, Reihenalgebra]
-- the iterative series approach -----------------------------------------
Here operations/functions are expressed as series; for instance the operation
of exponentiation as powerseries, such that
exp(x) = 1 + x + x^2/2 + ...
and then iteration
exp°2(x) = exp(exp(x)) = 1 + exp(x) + exp(x)^2/2! + ...
= K + Ax + Bx^2 + Cx^3 + ...
--- powerseries
The most common approach, see for instance []
b^x = 1 + log(b)x + log(b)^2 x^2/2! +...
(b,x)°1 = b^x
(b,x)°2 = b^(b^x) = 1 + log(b)b^x + log(b)^2 b^2x/2! + ...
(b,x)°h = (b,b^x)°(h-1)
The needed manipulations on powerseries are sometimes explicitely
expressed in matrix-notation [see links]
---functions defined by general series ( dirichlet-like series,..)
The needed calculations are performed using derivatives of the functions
in the places where for powerseries we use their coefficients
[see Bell-matrix, Carleman-matrix]
--- series on integer iterates of the function itself
(b,x)°h = A x + B*(b,x)°1 + C*(b,x)°2 + ...
[for instance see : binomial expansion]
The iterative series approach implies also a different view: one assumes
an "initial-value" x, to which the operation is applied iteration-times
which then gives a final value.
So in
(b,x)°h we have a ternary operation where we apply the exponentiation
using base b h-times to the initial-value x.
This paradigm is followed in [see: dynamical systems, iteration theory, ... ]
The reverse-engineering of this paradigm leads then to the redefinition of
the common "binary operators" using the iteration-form (we use a local,
one-way notation here)
b occurs h times
--------------------------------- -------------------
(+,b,x)°h : x + b + b + ... b : iterated addition
(+,b,x)°-h : x - b - b - ... b : iterated subtraction
(*,b,x)°h : x * b * b * ... b : iterated multiplication
(*,b,x)°-h : x / b / b / ... b : iterated division
(^,b,x)°h : b^ b ^ ... b^x : iterated exponentiation, right associative
(^,b,x)°-h : Log_b(Log_b(...Log_b(x)))
Though for iterated addition and multiplication no powerseries is required,
they may be consistently expressed the same way using matrix-operators on
formal powerseries, which perform addition, multiplication and exponentiation
in the ring of formal powerseries according to this scheme [see link]
The latter form of notation (or some convenient adaptions [see "notations"])
like exp_b°h(x) for (^,b,x)°h allows then to write the iteration-scheme
for any function, for instance
(dxp,b,x)°h = dxp_b°h(x) : where dxp_b(x):= b^x - 1
(sin,b,x)°h = sin_b°h(x) : where sin_b(x):= i*(b^(ix) - b^(-ix))/2
--- Connection between series-iteration and operator paradigms ------------
The common binary operators can then be seen as reduced forms
of the series-iteration approach
a + b == (+,b,a)°1
a + b + b
= a + 2*b == (+,b,a)°2
a - b == (+,b,a)°-1
b - b = 0 == (+,b,b)°-1
b + b = 2*b == (+,b,b)°1 = (+,b,(+,b,b)°-1))°2 = (+,b,0)°2
== (*,b,2)°1
a * b == (*,b,a)°1
a * b * b
= a*b^2 == (+,b,a)°2
a / b == (*,b,a)°-1
b / b = 1 == (*,b,b)°-1
b * b == (*,b,b)°1 = (*,b,(*,b,b)°-1))°2 = (*,b,1)°2
== (^,b,2)°1
b ^ a == (^,b,a)°1
b ^(b ^ a) == (^,b,a)°2
log_b(a) == (^,b,a)°-1
log_b(b) = 1 == (^,b,b)°-1
b ^ b == (^,b,b)°1 = (^,b,(^,b,b)°-1))°2 = (^,b,1)°2
Gottfried Helms, Kassel