This is the generating function for the fibonnaci sequence:
$$F(x) = \frac{x}{1 - x - x^{2} }$$
I need to get the sequence to find any element from this sequence directly. So I'm trying to find an $x$ coefficient:
$$F(x) = \frac{x}{1 - x - x^{2} } = \frac{x}{(1 - x y_{-})(1 - x y_{+}) } $$
1) I know quadratic equations can be written as $(x - x_{1})(x + x_{2})$, but here it's written differently - does this work for every quadratic equation?
$$F(x) = \frac{1}{(x_{1} - x_{2})}\left(\frac{1}{(1-xx_{1})} - \frac{1}{(1-xx_{2})}\right)$$
This step is understood.
$$F(x) = \frac{1}{\sqrt{5}}\left(\sum{x_{1}}^{n}{x}^{n} - \sum{x_{2}}^{n}{x}^{n}\right)$$
2) What happens here? How come that the epsilons have appeared here?
The next step is to take out the coefficients, it's not difficult so I understand this part.