This is Exercise 3.1.2 from Achim Klenke: »Probability Theory — A Comprehensive Course«.
Exercise: Give an example for two different probability generating functions that coincide at countably many points $x_i \in (0,1), i \in \mathbb{N}$.
I have no idea how this example should look like. To remind you, a generating function $\psi_X$ of a random variable $X$ is defined as \begin{align*} \psi_X&\colon [0, 1] \rightarrow [0,1] \\ z & \mapsto \sum_{n=0}^\infty \mathbf{P}[X=n]\, z^n \,. \end{align*}
I believe this exercise hasn't that much to do with probability theory. Just imagine the $\mathbf{P}[X=n]$ are numbers in $[0, 1]$, all adding up to 1, then it's a real analysis question.
Can anyone help me? Thank you!
Something that you might find helpful are Blaschke products. If a sequence of numbers $|z_n| < 1$ satisfy the condition $\sum_{n=0}(1-|z_n|) < \infty$ then there is a function analytic in the disc for which $f(z_n)=0$. In particular, the Blaschke product is such a function $$B(z)=\prod_{n=0}^\infty \frac{|z_n|}{z_n} \frac{z_n - z}{1-\bar{z_n}z}.$$
In particular if we choose the sequence $z=1-1/n^2$, this is such a sequence.
Now since $B(z_n)=0$ for each $n$, we have for any analytic function $f$: $f(z_n)B(z_n)=0$. Thus we have two functions $B(z)$ and $f(z)B(z)$ that vanish at the points $z_n$ and there are countably many of them.