Notation: $f^{(1)} (x) = f(x) \\ f^{(n)} (x) = f\left( f^{(n-1)}(x) \right)$
I'm looking for an explicit function, $f^{(n)} (x)$, where $f(x)$ is an arbitrary degree $2$ polynomial, or a nested quadratic.
I know there's a function for nested linear functions, $f(x) = mx+b \implies f^{(n)} (x) = m^n x + b\left( {m^n -1 \over m-1} \right)$, and $f(x) = x+b \implies f^{(n)} (x) = x+nb$.
I've been examining different kinds of quadratic functions, trying to find an explicit formula, but I can't catch the pattern. I know that $f(x) = ax^2 \implies f^{(n)}(x) = a^{2^n -1}x^{2^n}$, but I can't find a formula for $f(x) = x^2 +b$
I know it's a certain polynomial of the form $f(x) = x^2 +b \implies f^{(n)} (x) = x^{2^n} + \cdots + f^{(n)}(b)$, where the $\cdots$ are some chaotic behaving terms in the middle there, but it's the middle terms that I can't figure out.