Is $G(\alpha s)/G(\alpha)$ a probability generating function

Question

Suppose $G$ is a probability generating function. Let $\alpha\in [0,1]$, then is $\frac{G(\alpha s)}{G(\alpha)}$ necessarily a probability generating function?

Answer 1

Yes: a power series $\sum_{k=0}^\infty a_kx^k$ is a probability-generating function if and only if $\sum_{k=0}^\infty a_k=1$ and $\forall k, a_k\ge0$.

If $p(n)$ is the probability-mass function associated to $G$, then $\frac{p(n)\alpha^n}{G(\alpha)}$ is the one associated to $\frac{G(\alpha x)}{G(\alpha)}$. Notice that, since the radius of convergence of $G$ is a priori just $\ge1$, the whole thing might not work for $\alpha>1$.