Functional equation inspired by moment generating function

A user gave the following nice answer http://math.stackexchange.com/a/161584/5031

My question is that although it is clear $\log M_X(t)=Ct^2$ is a form that satisfies the condition $M_X^n(t/\sqrt{n})=M_X(t)$, how do you show no other form does?

My attempt: $\log M_X(t)$ can be written as $\alpha(t)t^2$ for some function $\alpha(t)$. Then $n\log M_X(t/\sqrt{n}) = n \alpha(t/\sqrt{n})t^2/n$. Letting $y=\sqrt{n}$, $\alpha(t)=\alpha(t/y)=\alpha(t/y^2)=\ldots$ so by continuity $\alpha(t)=\alpha(0)$, proving that $\alpha(t)$ is constant.

Any other ways of showing this?

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 Hmm, perhaps you could expand $M_X$ as a power series, and compare the coefficients of $M_X^n(t/\sqrt{n})$ and $M_X(t)$? – user1551 Nov 27 '12 at 15:34