Let for some base .

Then we demand that any tetration is a solution of the Abel equation

and .

Such a solution (even if analytic and strictly increasing) is generally not unique because for example the solution is also an analytic strictly increasing solution, by

and

For uniqueness it was Daniel's idea to consider the continuous iteration at a fixed point of . The continuous iteration of is derived from by

(0)

For such an iteration to be unique it suffices to demand the existence of the limit

(1) for the hyperbolic fixed point or

(2) for the parabolic fixed point (where is the index of the first non-zero coefficient in the development of at ).

So (2) is our uniqueness condition for and (1) is our uniqueness condition for .

Now I was thinking further that if is also analytic in , i.e. is an analytic function on (which has to be proved but is quite reasonable) then this function can be analytically extended to if there is no singularity at .

So this would mean we had a unique analytic (in and ) tetration under the condition (1) and (2), where is defined by (0).

Note, that we dont need a converging development of at the fixed point (which for , , is equivalent to the existence of a converging development of at 0). The analytic function is uniquely determined at least for and this suffices for our .

Then we demand that any tetration is a solution of the Abel equation

and .

Such a solution (even if analytic and strictly increasing) is generally not unique because for example the solution is also an analytic strictly increasing solution, by

and

For uniqueness it was Daniel's idea to consider the continuous iteration at a fixed point of . The continuous iteration of is derived from by

(0)

For such an iteration to be unique it suffices to demand the existence of the limit

(1) for the hyperbolic fixed point or

(2) for the parabolic fixed point (where is the index of the first non-zero coefficient in the development of at ).

So (2) is our uniqueness condition for and (1) is our uniqueness condition for .

Now I was thinking further that if is also analytic in , i.e. is an analytic function on (which has to be proved but is quite reasonable) then this function can be analytically extended to if there is no singularity at .

So this would mean we had a unique analytic (in and ) tetration under the condition (1) and (2), where is defined by (0).

Note, that we dont need a converging development of at the fixed point (which for , , is equivalent to the existence of a converging development of at 0). The analytic function is uniquely determined at least for and this suffices for our .