Consider the expansion of $e^{x}$, $$e^{x}=1+x+\frac{x^2}{2!}+\frac {x^3}{3!}+\cdots+\frac {x^n}{n!}+\cdots $$ Note that each term of the expansion on RHS, when divided by the closed form (LHS), gives the Poisson distribution, $Y \sim \text{Pois}(x)$ i.e. $$P(Y=n) = e^{-x}\cdot \frac {x^n}{n!}$$
Now consider the expansion of $\frac 1{1-x}$ (or, conversely, the infinite sum of a geometric progression), $$\frac 1{1-x}=1+x+x^2+\cdots+x^{n-1}+\cdots$$ Each term of the expansion on RHS, when divided by the closed form (LHS), gives the Geometric Distribution $Y \sim \text{Geo}(x)$, i.e. $$P(Y=n) = x^{n-1} (1-x)$$
Questions
Are there other similar discrete probability distributions where the probability of random variable $Y$ taking on value $n$ is given by e.g. the $n$-th term in the series expansion of a function divided by the closed form of the function?
Conversely, does this mean that every function with a series expansion has a corresponding discrete probability distribution? Which are the more commonly known functions and their corresponding probability distributions?
Obviously this will not work for series with alternating signs e.g. $\ln x$ or $\sin x$.
Any views or comments on this topic would be welcome. Thanks.
For any probability distribution $p$ supported on the nonnegative integers, the series $f(z) = \sum_{n=0}^\infty p(n) z^n$ defines the probability generating function of $p$. It is analytic on (at least) $|z| < 1$. If $X$ is a random variable with distribution $p$, $f(z) = \mathbb E[z^X]$.
If $f(z)$ is analytic in $|z|<1$ with $f(1) = 1$ and all its Maclaurin series coefficients are nonnegative, then $f(z)$ is the probability generating function of a probability distribution supported on the nonnegative integers.