TITLE: Recurrence relation: composition of a polynomial function
QUESTION [5 upvotes]: Let $f(x)=a+bx^2$. Define $f_n(x)$ to be the $n$-fold composition of $f$. That is
$$f_1(x)=f(x)$$
$$f_2(x)=f \circ f(x)$$
$$f_n(x)=f \circ f_{n-1}(x), n \ge 2$$
Is there a way to find a formula for $f_n$?
I tried to write down $f_2$, $f_3,\ldots$, but I don't see any pattern.

REPLY [4 votes]: After a simple change of variables, you will be iterating $z^2+c$ for some $c$.  There is a simple formula for the $n$th iterate when $c=0$ or $c=-2$.  But not otherwise.  
added 
Double-angle formula: $\cos 2\theta = 2\cos^2-1$, so if we write
$z=2\cos\theta$, then we get $f_1 = z^2-2 = 2\cos 2\theta$ ;
$f_2 = (z^2-2)^2-2 = 2\cos 4\theta$ ; and so on ... $f_n = 2\cos(2^n\theta)$.  If you like, put $\theta = \arccos(z/2)$ into these.