Generating Function for Linearly Recurrent Sequence

Theorem
Let $\sequence {a_n}$ be a linearly recurrent sequence defined as:
 * $a_n = \begin{cases}

b_n & : 1 \le n \le m \\ c_1 a_{n - 1} + c_2 a_{n - 2} + \cdots + c_m a_{n - m} & : n > m \end{cases}$

where:
 * $m \in \Z_{>0}$ is a (strictly) positive integer
 * $b_1, \ldots, b_m$ are constants.

Then the generating function for $\sequence {a_n}$ is of the form:
 * $\map G z = \dfrac {\map P z} {1 - c_1 z - c_2 z^2 - \cdots - c_m z^m}$

where $\map P z$ is a polynomial in $z$ given by $b_1 z + b_2 z^2 + \cdots + b_m z^m$.

Example
By this theorem, the generating function for $\sequence {F_n}$ is given by:


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