Generating function of a discrete variable

Question

Suppose that $G_X$ is the generating function of a discrete variable $X$. I have to determine the generating function of $nX$. I don’t know how to proceed.

Answer 1

We start with

$$ G_X(t) = \sum_{k=0}^\infty P(X = k)t^k $$

First let $Y = nX$, where $n \geq 1$ is an integer. Then $P(Y = nk) = P(X = k)$, which gives us

\begin{align} G_Y(t) & = \sum_{j=0}^\infty P(Y = j)t^j \\ & = \sum_{k=0}^\infty P(Y = nk)t^{nk} \qquad j = nk \\ & = \sum_{k=0}^\infty P(X = k)(t^n)^k \\ & = G_X(t^n) \end{align}

where we can make the identification $j = nk$ because no other possibilities for $j$ are permitted; $Y = nX$ can only take on values that are multiples of $n$.

Can you do something similar with $Z = n+X$?