# §5.15 Polygamma Functions

The functions $\mathop{\psi^{(n)}\/}\nolimits\!\left(z\right)$, $n=1,2,\dots$, are called the polygamma functions. In particular, $\mathop{\psi\/}\nolimits'\!\left(z\right)$ is the trigamma function; $\mathop{\psi\/}\nolimits''$, $\mathop{\psi^{(3)}\/}\nolimits$, $\mathop{\psi^{(4)}\/}\nolimits$ 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 $\mathop{\psi\/}\nolimits'\!\left(z\right)=\sum_{k=0}^{\infty}\frac{1}{(k+z)^{2% }},$ $z\neq 0,-1,-2,\dots$, Symbols: $\mathop{\psi\/}\nolimits\!\left(\NVar{z}\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable Permalink: http://dlmf.nist.gov/5.15.E1 Encodings: TeX, pMML, png See also: Annotations for 5.15
 5.15.2 $\mathop{\psi^{(n)}\/}\nolimits\!\left(1\right)=(-1)^{n+1}n!\mathop{\zeta\/}% \nolimits\!\left(n+1\right),$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $\mathop{\psi^{(\NVar{n})}\/}\nolimits\!\left(\NVar{z}\right)$: polygamma functions and $n$: nonnegative integer A&S Ref: 6.4.2 Referenced by: §5.15 Permalink: http://dlmf.nist.gov/5.15.E2 Encodings: TeX, pMML, png See also: Annotations for 5.15
 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 $\mathop{\psi\/}\nolimits'\!\left(n-\tfrac{1}{2}\right)=\tfrac{1}{2}\pi^{2}-4% \sum_{k=1}^{n-1}\frac{1}{(2k-1)^{2}},$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\psi\/}\nolimits\!\left(\NVar{z}\right)$: psi (or digamma) function, $n$: nonnegative integer and $k$: nonnegative integer A&S Ref: 6.4.5 Permalink: http://dlmf.nist.gov/5.15.E4 Encodings: TeX, pMML, png See also: Annotations for 5.15
 5.15.5 ${\mathop{\psi\/}\nolimits^{(n)}}\!\left(z+1\right)={\mathop{\psi\/}\nolimits^{% (n)}}\!\left(z\right)+(-1)^{n}n!z^{-n-1},$ Symbols: $\mathop{\psi\/}\nolimits\!\left(\NVar{z}\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable A&S Ref: 6.4.6 Permalink: http://dlmf.nist.gov/5.15.E5 Encodings: TeX, pMML, png See also: Annotations for 5.15
 5.15.6 ${\mathop{\psi\/}\nolimits^{(n)}}\!\left(1-z\right)+(-1)^{n-1}{\mathop{\psi\/}% \nolimits^{(n)}}\!\left(z\right)=(-1)^{n}\pi\frac{{\mathrm{d}}^{n}}{{\mathrm{d% }z}^{n}}\mathop{\cot\/}\nolimits\!\left(\pi z\right),$
 5.15.7 ${\mathop{\psi\/}\nolimits^{(n)}}\!\left(mz\right)=\frac{1}{m^{n+1}}\sum_{k=0}^% {m-1}{\mathop{\psi\/}\nolimits^{(n)}}\!\left(z+\frac{k}{m}\right).$ Symbols: $\mathop{\psi\/}\nolimits\!\left(\NVar{z}\right)$: psi (or digamma) function, $m$: nonnegative integer, $n$: nonnegative integer, $k$: nonnegative integer and $z$: complex variable A&S Ref: 6.4.8 Referenced by: §5.15 Permalink: http://dlmf.nist.gov/5.15.E7 Encodings: TeX, pMML, png See also: Annotations for 5.15

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

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

For $B_{2k}$ see §24.2(i).

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