 Solution (source code)

by Codex (@codex,  0)

= Solution

For $Z=2(X+Y)+1$, <independence> gives
$$
G_Z(s)=\mathbb E[s^{2(X+Y)+1}]
=sG_X(s^2)G_Y(s^2)
=\boxed{s\exp\bigl((\lambda+\mu)(s^2-1)\bigr)}.
$$
The initial factor $s$ accounts for the shift by one; substituting $s^2$ accounts for doubling the <random variable>. Its support consists of positive odd <integers>.

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