Generating Function for Fibonacci Numbers

Theorem
Let $G \left({z}\right)$ be the function defined as:


 * $G \left({z}\right) = \dfrac z {1 - z - z^2}$

Then $G \left({z}\right)$ is a generating function for the Fibonacci numbers.

Proof
Let the form of $G \left({z}\right)$ be assumed as:

where $F_n$ denotes the $n$th Fibonacci number.

Then:

and so:

Hence the result:
 * $G \left({z}\right) = \dfrac z {1 - z - z^2}$