Generating Function for Even Terms of Sequence

Theorem

Let $\map G z$ be the generating function for the sequence $\sequence {a_n}$.

Consider the subsequence $\sequence {b_n} := \tuple {a_0, a_2, a_4, \ldots}$

Then the generating function for $\sequence {b_n}$ is:

$\dfrac 1 2 \paren {\map G z + \map G {-z} }$

$\ds \map G z$

$=$

$\ds \sum_{n \mathop \ge 0} a_n z^n$

$\ds $

$=$

$\ds \sum_{r \mathop \ge 0} a_{2 r} z^{2 r} + \sum_{r \mathop \ge 0} a_{2 r + 1} z^{2 r + 1}$

separating out odd and even integers

$\ds \map G {-z}$

$=$

$\ds \sum_{n \mathop \ge 0} a_n \paren {-z}^n$

$\ds $

$=$

$\ds \sum_{n \mathop \ge 0} \paren {-1}^n a_n z^n$

$\ds $

$=$

$\ds \sum_{r \mathop \ge 0} \paren {-1}^{2 r} a_{2 r} z^{2 r} + \sum_{r \mathop \ge 0} \paren {-1}^{2 r + 1} a_{2 r + 1} z^{2 r + 1}$

separating out odd and even integers

$\ds $

$=$

$\ds \sum_{r \mathop \ge 0} a_{2 r} z^{2 r} - \sum_{r \mathop \ge 0} a_{2 r + 1} z^{2 r + 1}$

$\ds \leadsto \ \ $

$\ds \map G z + \map G {-z}$

$=$

$\ds 2 \sum_{r \mathop \ge 0} a_{2 r} z^{2 r}$

Hence the result.

$\blacksquare$