Suppose the $Z$-transform of $x(n)$ is $X(z)$. What is the $Z$-transform of $x(2n)$?

Question

Suppose the $Z$-transform of $x(n)$ is $X(z)$. What is the $Z$-transform of $x(2n)$?

Here is my thought process:

$X_2(z) =\displaystyle\sum_{- \infty}^{\infty}x_2(n)z^{-n}=\sum_{- \infty}^{\infty}x(2n)z^{-n}$.

But at this point I am unclear how to go about it.

Answer 1

Try $$\frac12 (X(\sqrt z) + X(-\sqrt z)) $$