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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?

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I don't see what's quadratic about your map, except for the deceptive notation in the first equation. – Raskolnikov Aug 9 '12 at 5:09
@Raskolnikov: I didn't intend to be deceptive. I'm talking about "quadratic maps", which are a specific topic, and not just maps that are quadratic, or quadratic equations that are maps. They are defined recursively in a quadratic fashion. I've been trying to find more information about them. There is some literature under "dyanamic systems", if that helps. – Matt Groff Aug 9 '12 at 5:57
I don't see where you're getting a power series. $a_1$ is of degree 2, $a_2$ of degree 4, $a_3$ of degree 8 in $a_0$, etc., but that's a sequence of polynomials - where's the power series? And what do you have in mind when you ask for a generating function? – Gerry Myerson Aug 9 '12 at 6:03
@GerryMyerson: Sorry, changed the question to say I'm using a generating function. There's no power series. I have in mind a closed form in the strictest sense, I believe. Just a function involving the variable $x$ and very basic/elementary arithmetic - not even summations or integrations. The only other things present would be the exact expression for $a_0$. I would like to know whether or not this could be useful. I have asked a few related questions lately. – Matt Groff Aug 9 '12 at 6:16
There is still a power series, you have written $A(x)=$ and what's on the right side of that equation is a power series. But I think what fooled me is your expansion of $a_2$, which made me misunderstand what you're doing. Anyway, what do you mean by "get" a generating function? You can certainly write down the generating function for any sequence whatsoever, quadratic or otherwise. – Gerry Myerson Aug 9 '12 at 7:02
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