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non-natural operation ranks
#1
Some basic thoughts

If we want to summarize our operation sequence [n] we can perhaps write:

, where


,

So we have a certain operator that assigns to a given function the function , this operator may be based on the natural Abel method, the diagonalization method, or the regular Abel method (with restrictions of ) and we can write

, moreover


.

Note: I original found it more appropriate to start with the addition as 0th instead of the 1st operation. However I adapted to the already established nomenclature. If we would stick to my original counting (addition as 0th operation) we had the better looking formula

As we have some methods for real functions to switch from to non-natural iterations maybe there are also methods for the operator to compute which then gives definition for real and complex iteration ranks:

.

In the moment however it is even unclear how to generally express such operators which map powerseries.

At least we can determine that is the inverse operator, i.e. is a function such that , i.e. . So is unique (independent of the method of and independent on the initial condition).

If we compute from this view point we get also:
and

...
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#2
bo198214 Wrote:Some basic thoughts

If we want to summarize our operation sequence [n] we can perhaps write:

, where


,

I was thinking of a function like:

so that for every combination of 3 at least complex variables we get one as a result. The type of number that will be the result is not obvious - there has been a discussion that new number types may appear.
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#3
Ivars Wrote:I was thinking of a function like:

so that for every combination of 3 at least complex variables we get one as a result.

Exactly, everybody thinks of such a function. Thatswhy I showed a direction how one can obtain such a function, i.e. by:

.

where is not a function, but an operator that maps functions to functions. The idea is to apply non-natural iteration to such an operator in a similar way as we apply non-natural iteration to a function. If we can compute non-natural iterates of functions via matrices, perhaps we can compute non-natural iterates of operators by the by Andrew mentioned tensors, who knows.
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#4
bo198214 Wrote:.

I must say, this is so much more beautiful than my "hyper-exponential" section of the most recent Reference (I mistakenly called "FAQ"). However, it seems that this suffers from similar vagueness of the "iterative logarithm" in that it has both a function-parameter and a value-parameter. Can regular iteration even be performed on this?

Andrew Robbins
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