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

...

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

...