 Solution

Usingwe 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 $n\ge 1$, $z=-n$ is removable when $p$ is one of $1,2,\dots ,n$, and otherwise is a simple pole.