When to use Moment generating functions compared to probability generating functions Is it always the pgf for discrete distributions and mgf for continuous distributions? 
 A: Technically, for a random variable $X$, the probability generating function (pgf)
$$
P_X(x)=\sum_{n=0}^\infty p_n x^n
\quad\text{where}\quad p_n=\Pr[X=n]
$$
only applies to discrete distributions on the non-negative integers (although you might extend that to the distributions on integers with some limitations).
The moment generating function (mgf)
$$
M_X(s)=\text{E}\left[e^{sX}\right]
=\sum_{n=0}^\infty \frac{M_nx^n}{n!}
\quad\text{where}\quad M_n=\text{E}[X^n]
$$
always applies, for discrete as well as continuous distributions. (However, it need not converge for non-zero values of $s$).
Which to use depends on your need, and is sometimes just a matter of preference as they are related by $M_X(s)=P_X(e^s)$.
However, in many cases you can compute the momenta $M_n$ more easily than the probabilities $p_n$. Although $M_X(s)=P_X(e^s)$ holds true and you can easily obtain the mgf from the pgf, it may be tricky to obtain the probabilities from the momenta.
Beware that the one-to-one correspondence between $M_X(s)$ and $P_X(x)$ requires that $M_X(s)$ is defined (ie converges) for $s$ in some open interval around zero. Otherwise, you can find different probability distributions with the same momenta.  
