PDF for Moment Generating Function $(1 +\beta t)^{-\alpha}$ when $\beta > 0$

Question

I'm trying to find the PDF for the following MGF.

$$(1 + t/2)^{-3/2}$$

I already know that $(1 - \beta t)^{-\alpha}$ when $\beta > 0$ is an MGF for the Gamma distribution $\Gamma(\alpha, \beta$).

However, the above formula since it is clearly not Gamma distributed, due to its $\beta$'s sign.

Is there anyone to give me an hint to find the MGF for the above formula, or, more generally,

$(1 +\beta t)^{-\alpha}$ when $\beta > 0$?

Answer 1

Hint:

If $X$ has moment generating function $$(1 - \beta t)^{-\alpha}$$

then $Y=-X$ has moment generating function $$(1 +\beta t)^{-\alpha}$$