\begin{definition}[Definition:Polygamma Function]
The $n$th '''polygamma function''', $\psi_n$, is defined, for $z \in \C \setminus \Z_{\le 0}$, by the $n$th derivative of the digamma function: 
:$\map {\psi_n} z = \dfrac {\d^n} {\d z^n} \map \psi z$
where $\psi$ is the digamma function.
\end{definition}