Linear Combination of Generating Functions

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

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

Then $\alpha G \left({z}\right) + \beta H \left({z}\right)$ is the generating function for the sequence $\left\langle{\alpha a_n + \beta b_n}\right\rangle$.

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

Let $G \left({z}\right)$ and $H \left({z}\right)$ converge to $x$ and $y$ respectively for some $z_0 \in \R_{>0}$.

Then from Linear Combination of Convergent Series:
 * $\displaystyle \sum_{n \mathop \ge 0} \left({\alpha a_n + \beta b_n}\right) z^n = \alpha \sum_{n \mathop \ge 0} a_n z^n + \beta \displaystyle \sum_{n \mathop \ge 0} b_n z^n$

Hence the result.