Given that $F(x)=\sum_{n=0}^\infty F_nx^n= \frac{x}{1-x-x^2}$, where $F_n(x)$ is the $n^{th}$ term of the Fibonacci series, and $F(x)$ is the generating function associated to it, find the generating function associated to $F_{2n}$
I know that $F_{2n}=F^2_n+2F_nF_{n-1}$ but this doesn't seem to help much. How can I do this?
Best Answer
We want to compute $$\sum_{n\geqslant 0} \ F_{2n} x^n$$
Recall the fact that $$F(x)=\sum_{n\geqslant 0} \ F_{n} x^n=F_0+F_1x+F_2x^2+\cdots=\frac{x}{1-x-x^2}$$ Now, $$\frac{1}{2}(F(x)+F(-x))=F_0+F_2x^2+F_4x^4+\cdots=\sum_{n\geqslant 0} F_{2n}x^{2n}$$ Therefore, we have $$\sum_{n\geqslant 0} \ F_{2n} x^n=F_0+F_2x+F_4x^2+\cdots=\frac{1}{2}\left(F(\sqrt{x})+F(-\sqrt{x})\right)=\frac{x}{1-3x+x^2}$$ The problem is solved.