Product of Generating Functions

Theorem
Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$.

Let $H \left({z}\right)$ be the generating function for the sequence $\left\langle{b_n}\right\rangle$.

Then $G \left({z}\right) H \left({z}\right)$ is the generating function for the sequence $\left\langle{c_n}\right\rangle$, where:


 * $\forall n \in \Z_{\ge 0}: c_n = \displaystyle \sum_{k \mathop = 0}^n a_k b_{n - k}$

Proof
By definition of generating function:


 * $G \left({z}\right) = \displaystyle \sum_{n \mathop \ge 0} a_n z^n$
 * $H \left({z}\right) = \displaystyle \sum_{n \mathop \ge 0} b_n z^n$

Then:


 * $\begin{array}{c|ccccc}

\times & b_0 & b_1 z & b_2 z^2 & b_3 z^3 & \cdots \\ \hline a_0 & a_0 b_0 & a_0 b_1 z & a_0 b_2 z^2 & a_0 b_3 z^3 & \cdots \\ a_1 z & a_1 b_0 z & a_1 b_1 z^2 & a_1 b_2 z^3 & a_1 b_3 z^4 & \cdots \\ a_2 z^2 & a_2 b_0 z^2 & a_2 b_1 z^3 & a_2 b_2 z^4 & a_2 b_3 z^5 & \cdots \\ a_3 z^3 & a_3 b_0 z^3 & a_3 b_1 z^4 & a_3 b_2 z^5 & a_3 b_3 z^6 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array}$

The result follows by definition of generating function.