Probability generating functions for a linear change of variable? I am struggling to understand, why, in textbook exercise on probability generating functions it says that if $X$ is a random variable (rv) with probability generating function (pgf)  $G_{X}(s)$, then for rv $Z = X + k$ for positive integer $k$ the pgf is given by $$G_{Z}(s) = s^{k}G_{X}(s).$$
Surely if we assume that the rv $X$ takes integer values $0,1,2,,,$ with probability $p_{0}, p_{1}, p_{2},...$then  the rv $Z$ takes integer values $k, k+1, k+2,...$ with corresponding probability $p_{0}, p_{1}, p_{2},...$, and thus the corresponding generating functions should be identical? The only thing that changes is the integer starting position (it is shifted by $k$) for the distribution of rv?

Let $G_X(s)$ be the probability generating function (PGF) of the random variable $X$.  Then if $Z = X + k$ for some fixed $k$, we simply have $$G_Z(s) = \operatorname{E}[s^Z] = \operatorname{E}[s^{X + k}] = \operatorname{E}[s^k s^X] = s^k \operatorname{E}[s^X] = s^k G_X(s).$$
