I'm trying to find a Taylor series for g^n, the nth iterate of g, centered at a fixed point p of g. I know the first few terms: if g^n(x)=a_0 + a_1 (x-p) + a_2 (x-p)^2 + ..., then let s=g'(p) a_0=p a_1=s^n a_2=[s^(n-1) (s^n - 1) g''(p)] / (2(s-1)). It's safe to say that a_3 is as complex as a_2 is compared to a_1; it's not pretty. I have a feeling that Faá di Bruno's formula (http://mathworld.wolfram.com/FaadiBrunosFormula.html) may be involved but I would prefer to avoid that. One of my goals is to find a formula in which I can let n be something other than an integer to at least approximate some fractional iterates of a function. Does anyone know a formula for the kth derivative of the nth iterate of g (assuming its sufficiently well behaved)? Thanks.