The functions ${\psi }^{(n)}\left(z\right)$, $n=1,2,\mathrm{\dots }$, are called the polygamma functions. In particular, ${\psi }^{\prime }\left(z\right)$ is the trigamma function; ${\psi }^{\prime \prime }$, ${\psi }^{(3)}$, ${\psi }^{(4)}$ are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. This includes asymptotic expansions: compare §§2.1(ii)–2.1(iii).

In (5.15.2)–(5.15.7) $n,m=1,2,3,\mathrm{\dots }$, and for $\zeta \left(n+1\right)$ see §25.6(i).

As $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \pi -\delta$

For $B_{2k}$ see §24.2(i).

For continued fractions for ${\psi }^{\prime }\left(z\right)$ and ${\psi }^{\prime \prime }\left(z\right)$ see Cuyt et al. (2008, pp. 231–238).