I'm trying to find a Taylor series for g^n, the nth iterate of g, centered at a fixed point p of g.(adsbygoogle = window.adsbygoogle || []).push({});

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 FaĆ” 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.

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# Kth derivative of the nth iterate

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