Polygamma function

Summary

The polygamma function of order $m$ is the $m$-th derivative of the digamma function, so the $(m+1)$-th derivative of $\log \Gamma$. For $m \ge 1$ it is a series of inverse powers, closely tied to the Hurwitz zeta function. Its values at $1$ are, up to sign and a factorial, values $\zeta(m+1)$ of the Riemann zeta function. The case $m = 1$ is the trigamma function.