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Suppose $\{a_n\}$ is a sequence with moment generating function $A(z)=\sum_{k \ge 0} a_kz^k$.

Can a sequence $\{b_n\}$ with $b_n \neq a_n$ for at least one $n\in \mathbb N$ have the same moment generating function as $\{a_n\}$, that is $B(z)=A(z) \ \forall z$.

This is equivalent to: Is the moment generating function for a sequence $\{a_n\}$ unique ?

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    $\begingroup$ Afaik, it's called ordinary generation function in this context. MGF are a slightly different thing, conceptually. Anyway, is there a particular reason you did not post this on Mathematics? Oh wait, you did! Please don't do that. I'm closing it here because it's a better fit for Mathematics. $\endgroup$ – Raphael Mar 22 '14 at 22:57