Continuous iteration

[andydude]

The oldest reference I know of is Koch [2], which according to Bennett [1], was the first to consider using Bell-Carleman matrices to derive the "regular" formal series of a continuous iteration of a function. But since we're talking about different things, I can see how your reference can be older, since you're talking about the series themselves, not finding them through matrices. Bennett also mentions Koenigs, so he obviously knew of him.

[1] Albert A. Bennett, The Iteration of Functions of one Variable, The Annals of Mathematics, 2nd Ser., Vol. 17, No. 1, pp. 23-60, (Sep. 1915).
[2] H. v. Koch, Bil. t. Sv. Vet. Ak. Hand. 1, Math. 25, m\'em. no. 5, pp. 1-24, (1900).

[bo198214]

since you're talking about the series themselves, not finding them through matrices.

Yes, Koenigs method is an approximation method for the function outside the fixed point. He does not develop the coefficients of the iterated function (I think).

However that iterative roots are unique (in the real hyperbolic case) for formal power series must have been common knowledge for a quite long time, as it is easy to derive a recursive formula from the Faá di Bruno formula (without knowing about the matrix correspondence). And the steps from roots to fractional to continuous iteration are little ones.