If I have a gamma random variable $X$ with PDF
$$ f(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha - 1}e^{-\beta x},\quad x > 0,\,\alpha>0,\,\beta > 0,$$
then I know $X$ has MGF
$$F(z) = \mathbb{E}\left[e^{zX}\right] = \left(1 - \frac{z}{\beta}\right)^{-\alpha},\quad z < \beta.$$
If I start with $F$, and I allow $\alpha \leq 0$, then $F$ is no longer the MGF of a gamma random variable. Is this altered $F$ the MGF of some other random variable? If so, are these studied/have a name?
If $\alpha=0$ then $F(z)=1$. This is the MGF of degenerate at zero distribution $X=0$ a.s. Note that for this distribution MGF exists for any $z$, not only for $z<\beta$.
If $\alpha<0$, then $F(z)=\left(1-\frac{z}{\beta}\right)^{-\alpha}$ ($z<\beta$) is not MGF at all. Indeed, consider this function as MGF and find contradiction. Denote $-\alpha$ by $a$.
Look at the first two moments of r.v. $X$ with MGF $F(z)=\left(1-\frac{z}{\beta}\right)^{a}$, $z<\beta$. $$ \mathbb E[X] = \frac{dF}{dz}\Big|_{z=0} = -\frac{a}{\beta}\left(1-\frac{z}{\beta}\right)^{a-1}\Big|_{z=0} = -\frac{a}{\beta}. $$ $$ \mathbb E[X^2] = \frac{d^2F}{dz^2}\Big|_{z=0} = \frac{a(a-1)}{\beta^2}\left(1-\frac{z}{\beta}\right)^{a-2}\Big|_{z=0} = \frac{a(a-1)}{\beta^2}. $$ Find variance: $$ \mathop{Var}(X) = \mathbb E[X^2] - \left(\mathop{\mathbb E}[X]\right)^2 = \frac{a(a-1)}{\beta^2} - \frac{a^2}{\beta^2} = -\frac{a}{\beta^2}<0. $$ This means that $F(z)$ is not a MGF of any distribution.