I wonder about Woon's expansion as you gave it:

What is the parameter ???

I mean the fractional iteration of a formal power series f is unique as long as and and the above derivation would contradict this uniqueness.

Let me show the uniqueness of fractional iteration for the example of a compositional square root.

Take an arbitrary formal powerseries and look for the compositional square root , i.e. a formal powerseries such that .

For this we need a formula for the composition of two formal powerseries. If we innocently start computing it:

we realize that we need the -th power of the powerseries g, i.e. at least a formula for multiplication:

If we generalize this to the multiplication of an arbitrary number of series we get for the coefficients of the -th power

Then we put this into our composition computation

If we now assume that then the sum is finite because for at least one , which causes and hence the whole product .

This gives

Now lets go back to the solution of . By the above formula the coefficients must satisfy the following equations

For , the composition formula reduces to . If we assume then is uniquely determined as

.

For the are all smaller than m, except for n=1, because otherwise all other . So the only terms containing that can occur on the left side is and . And this gives us a mean to recursively define , i.e. by

.

This reasoning can be extended to arbitrary natural exponents, and shows us that in the domain of formal powerseries f with and the fractional iteration is unique.

What is the parameter ???

I mean the fractional iteration of a formal power series f is unique as long as and and the above derivation would contradict this uniqueness.

Let me show the uniqueness of fractional iteration for the example of a compositional square root.

Take an arbitrary formal powerseries and look for the compositional square root , i.e. a formal powerseries such that .

For this we need a formula for the composition of two formal powerseries. If we innocently start computing it:

we realize that we need the -th power of the powerseries g, i.e. at least a formula for multiplication:

If we generalize this to the multiplication of an arbitrary number of series we get for the coefficients of the -th power

Then we put this into our composition computation

If we now assume that then the sum is finite because for at least one , which causes and hence the whole product .

This gives

Now lets go back to the solution of . By the above formula the coefficients must satisfy the following equations

For , the composition formula reduces to . If we assume then is uniquely determined as

.

For the are all smaller than m, except for n=1, because otherwise all other . So the only terms containing that can occur on the left side is and . And this gives us a mean to recursively define , i.e. by

.

This reasoning can be extended to arbitrary natural exponents, and shows us that in the domain of formal powerseries f with and the fractional iteration is unique.