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Given a function $G(z)$, what are the sufficient and necessary conditions for being a $G(z)$ to be a probability generating function.

Few necessary condition which I know of are

  1. $G(1) = 1$
  2. All the coefficients of $z^n$ must be positive.

Thanks in advance!

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Sufficient and necessary condition for $G$ to be a generating function is that $G$ is smooth on $(-1,1)$ with every derivative $G^{(n)}(0)$ at $0$ nonnegative and $\sum\limits_{n=0}^\infty\frac1{n!}G^{(n)}(0)=1$.

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