Polygamma Function in terms of Hurwitz Zeta Function

Theorem

 * $\displaystyle \map {\psi_n} z = \paren {-1}^{n + 1} \map \Gamma {n + 1} \map \zeta {n + 1, z}$

where:
 * $\psi_n$ is the polygamma function
 * $\Gamma$ is the gamma function
 * $\zeta$ is the Hurwitz zeta function
 * $z$ is a complex number.
 * $n \in \Z_{\ge 1}$.