Let $F_n$ be the Fibonacci number, with $F_0 = 1, F_1 = 1$. Prove the identity $2F_n = F_{n+1} + F_{n−2}$ by using generating functions.

Question

Let Fn be the Fibonacci number, with $F_0 = 1, F_1 = 1$. Prove the identity $2F_n = F_{n+1} + F_{n−2}$ by using generating functions.

I know the generating function for the shifted Fibonacci Sequence, $F_n$ with $F_0=1$ and $F_1=1$ but not sure how to incorporate the $2$.

Answer 1

You may prove it by using the direct formula for Fibonacci numbers $$F_k=\frac{a^k-b^k}{\sqrt{5}}, a+b=1, ab=-1,a^2=a+1, b^2=b+1$$, Then $$F_{n+1}+F_{n-2}= \frac{a^{n+1}-b^{n+1}}{\sqrt{5}}+\frac{a^{n-2}-b^{n-2}}{\sqrt{5}}$$ $$=\frac{1}{\sqrt{5}}\left(a^n(a+\frac{1}{a^2})-b^n(b+\frac{1}{b^2}) \right)$$ note that $a+1/a^2=a+b^2=a+b+1=2$, then $$F_{n+1}+F_{n-2}= \frac{2}{\sqrt{5}}[a^n-b^n]=2 F_n$$