Hadamard's product of Fibonacci generating functions.

Question

$F(s) = \frac{1}{1-s-s^2}=\sum_{n\geq0}F_ns^n$. I want to calculate $F(s) \circ F(s) = \sum_{n\geq0}F_{n}^2s^n$. I have tried using Binet"s formula, but problem remains unsolved.

Answer 1

Square the Binet formula to get $$F_n^2=\frac{\phi^{2n}-2(-1)^n+\phi^{-2n}}{5},$$where $\phi=(1+\sqrt5)/2$. Now the series splits into three geometric series, where the common ratios are respectively $\phi^2s$, $-s$, and $\phi^{-2}s$.