Solution
= Solution
Using
$$
B(p,z)=\frac{\Gamma(p)\Gamma(z)}{\Gamma(p+z)},
$$
we see that $z=-n$ is generally a simple pole, inherited from $\Gamma(z)$. A cancellation occurs exactly when $p-n$ is a nonpositive integer. Thus $z=0$ is always a simple pole; for $n\geq1$, $z=-n$ is removable when $p$ is one of $1,2,\ldots,n$, and otherwise is a simple pole.