# §5.15 Polygamma Functions

The functions $\mathop{\psi^{(n)}\/}\nolimits\!\left(z\right)$, $n=1,2,\dots$, are called the polygamma functions. In particular, ${\mathop{\psi\/}\nolimits^{\prime}}\!\left(z\right)$ is the trigamma function; ${\mathop{\psi\/}\nolimits^{\prime\prime}}$, $\mathop{\psi^{(3)}\/}\nolimits$, $\mathop{\psi^{(4)}\/}\nolimits$ 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) $n,m=1,2,3,\dots$, and for $\mathop{\zeta\/}\nolimits\!\left(n+1\right)$ see §25.6(i).

 5.15.1 ${\mathop{\psi\/}\nolimits^{\prime}}\!\left(z\right)=\sum_{k=0}^{\infty}\frac{1% }{(k+z)^{2}},$ $z\neq 0,-1,-2,\dots$,
 5.15.2 $\mathop{\psi^{(n)}\/}\nolimits\!\left(1\right)=(-1)^{n+1}n!\mathop{\zeta\/}% \nolimits\!\left(n+1\right),$
 5.15.3 $\mathop{\psi^{(n)}\/}\nolimits\!\left(\tfrac{1}{2}\right)=(-1)^{n+1}n!(2^{n+1}% -1)\mathop{\zeta\/}\nolimits\!\left(n+1\right),$
 5.15.4 ${\mathop{\psi\/}\nolimits^{\prime}}\!\left(n-\tfrac{1}{2}\right)=\tfrac{1}{2}% \pi^{2}-4\sum_{k=1}^{n-1}\frac{1}{(2k-1)^{2}},$
 5.15.5 ${\mathop{\psi\/}\nolimits^{(n)}}\!\left(z+1\right)={\mathop{\psi\/}\nolimits^{% (n)}}\!\left(z\right)+(-1)^{n}n!z^{-n-1},$
 5.15.6 ${\mathop{\psi\/}\nolimits^{(n)}}\!\left(1-z\right)+(-1)^{n-1}{\mathop{\psi\/}% \nolimits^{(n)}}\!\left(z\right)=(-1)^{n}\pi\frac{{d}^{n}}{{dz}^{n}}\mathop{% \cot\/}\nolimits\!\left(\pi z\right),$
 5.15.7 ${\mathop{\psi\/}\nolimits^{(n)}}\!\left(mz\right)=\frac{1}{m^{n+1}}\sum_{k=0}^% {m-1}{\mathop{\psi\/}\nolimits^{(n)}}\!\left(z+\frac{k}{m}\right).$

As $z\to\infty$ in $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta\;(<\pi)$

 5.15.8 ${\mathop{\psi\/}\nolimits^{\prime}}\!\left(z\right)\sim\frac{1}{z}+\frac{1}{2z% ^{2}}+\sum_{k=1}^{\infty}\frac{\mathop{B_{2k}\/}\nolimits}{z^{2k+1}}.$

For $\mathop{B_{2k}\/}\nolimits$ see §24.2(i).

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