If we suppose that we can get a generating function for any "quadratic map (as in dynamical systems)", what are the mathematical applications? Also, what are the "real world" applications of this? Would this, for instance, allow new computations that were previously unobtainable?
MORE DETAILED EXPLANATION
To show the problem, we start with an equation, which is our "quadratic map": $a_{n+1} = A(a_n)^2 + B(a_n) + C$
Then we map it out into a generating function: $A(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n + \dots$
So, for instance, $a_1 = A(a_0)^2 + B(a_0) + C$ $a_2 = A\left(A(a_0)^2 + B(a_0) + C \right)^2 + B\left(A(a_0)^2 + B(a_0) + C \right) + C$ $= A^2(a_0)^4+2AB(a_0)^3+(2AC+B^2+A)(a_0)^2+(2BC + B)a_0+(C^2+2C)$ $\dots$
Now we can suppose, for instance, that we know a very simple formula for $A(x)$. In other words, $a_n$ may have a very complicated formula in terms of $a_0$, but the formula for $A(x)$ could be relatively simple in some cases. How can the $A(x)$ simplification be used to advantage?