Iterated polynomials

[JmsNxn]

I've seen some of you talking about half-iterates of the sin function and I can only guess this is from a general formulae about partial iterates of polynomial functions.

the idea is too take half iterates of formal powerseries (not just polynomials).
There is a formula for the composition of formal powerseries and you solve then something like \( f\circ f = F \), comparing and hence determining the coefficients of the formal powerseries. This works only for \( F(0)=0 \), i.e. \( F \) having a fixed point at 0.

In certain cases (e.g. \( F'(0)=1 \) so called parabolic fixed point at 0, e.g. \( e^x-1 \) and \( \sin(x) \)) the resulting powerseries is however not converging. Then you need to take resort in other methods. Perhaps first look at the FAQ, to find in the "General Discussion and Questions". And next take a look at
http://mathoverflow.net/questions/4347/f...5227#45227

I also read your other post, it looks very interesting, however for the next weeks I dont have time to dive into. I hope some other forum members reply.

Thank you for the link. I said polynomials because I thought there would be a general formula which would extend to power series.

I hope someone replies to my other post as well.