How can I show Fibonacci identity?

Question

How can I Show a Fibonacci identity: $$\sum_{k=0}^{n}F_{k}F_{n-k}=\sum_{k=0}^{n}(k+1)F_{k+1}(-2)^{n-k}$$ Can anyone help me, i have no idea.

Answer 1

You may use the ordinary generating function of Fibonacci numbers: $$ \sum_{n\geq 0}F_n x^n = \frac{x}{1-x-x^2} \tag{1}.$$ Letting $[x^m]\,g(x)$ be the coefficient of $x^m$ in the Maclaurin series of $g(x)$, we have $$ \sum_{k=0}^{n} F_k F_{n-k} = [x^n]\left(\frac{x}{1-x-x^2}\right)^2\tag{2} $$ and $$ \sum_{k=0}^{n}(k+1)F_{k+1}(-2)^{n-k} = [x^n]\frac{1}{1+2x}\left(\frac{x}{1-x-x^2}\right)' \tag{3}$$ so the LHS and RHS are different things.