Generating Function for Fibonacci Numbers

Theorem

Let $\map G z$ be the function defined as:

$\map G z = \dfrac z {1 - z - z^2}$

Let the form of $\map G z$ be assumed as:

$\ds \map G z$

$=$

$\ds \sum_{k \mathop \ge 0} F_k z^k$

$\ds $

$=$

$\ds F_0 + F_1 z + F_2 z^2 + F_3 z^3 + F_4 z^4 + \cdots$

$\ds $

$=$

$\ds 0 + z + z^2 + 2 z^3 + 3 z^4 + \cdots$

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

Then:

$\ds z \map G z$

$=$

$\ds F_0 z + F_1 z^2 + F_2 z^3 + F_3 z^4 + F_4 z^5 + \cdots$

$\ds z^2 \map G z$

$=$

$\ds F_0 z^2 + F_1 z^3 + F_2 z^4 + F_3 z^5 + F_4 z^6 + \cdots$

and so:

$\ds \paren {1 - z - z^2} \map G z$

$=$

$\ds F_0 + \paren {F_1 - F_0} z + \paren {F_2 - F_1 - F_0} z^2 + \paren {F_3 - F_2 - F_1} z^3 + \cdots$

$\ds $

$=$

$\ds F_0 + \paren {F_1 - F_0} z$

Definition of Fibonacci Number: $F_n = F_{n - 1} + F_{n - 2}$

$\ds $

$=$

$\ds z$

Definition of Fibonacci Number: $F_0 = 0, F_1 = 1$

Hence the result:

$\map G z = \dfrac z {1 - z - z^2}$

$\blacksquare$