# §5.15 Polygamma Functions

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

 5.15.1 $\psi'\left(z\right)=\sum_{k=0}^{\infty}\frac{1}{(k+z)^{2}},$ $z\neq 0,-1,-2,\dots$, ⓘ Symbols: $\psi\left(\NVar{z}\right)$: 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 and Ch.5
 5.15.2 $\psi^{(n)}\left(1\right)=(-1)^{n+1}n!\zeta\left(n+1\right),$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $\psi^{(\NVar{n})}\left(\NVar{z}\right)$: 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 and Ch.5
 5.15.3 $\psi^{(n)}\left(\tfrac{1}{2}\right)=(-1)^{n+1}n!(2^{n+1}-1)\zeta\left(n+1% \right),$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $\psi^{(\NVar{n})}\left(\NVar{z}\right)$: polygamma functions and $n$: nonnegative integer A&S Ref: 6.4.4 Permalink: http://dlmf.nist.gov/5.15.E3 Encodings: TeX, pMML, png See also: Annotations for §5.15 and Ch.5
 5.15.4 $\psi'\left(n-\tfrac{1}{2}\right)=\tfrac{1}{2}\pi^{2}-4\sum_{k=1}^{n-1}\frac{1}% {(2k-1)^{2}},$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$: 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 and Ch.5
 5.15.5 ${\psi}^{(n)}\left(z+1\right)={\psi}^{(n)}\left(z\right)+(-1)^{n}n!z^{-n-1},$ ⓘ Symbols: $\psi\left(\NVar{z}\right)$: 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 and Ch.5
 5.15.6 ${\psi}^{(n)}\left(1-z\right)+(-1)^{n-1}{\psi}^{(n)}\left(z\right)=(-1)^{n}\pi% \frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\cot\left(\pi z\right),$
 5.15.7 ${\psi}^{(n)}\left(mz\right)=\frac{1}{m^{n+1}}\sum_{k=0}^{m-1}{\psi}^{(n)}\left% (z+\frac{k}{m}\right).$ ⓘ Symbols: $\psi\left(\NVar{z}\right)$: 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 and Ch.5

As $z\to\infty$ in $|\operatorname{ph}z|\leq\pi-\delta$

 5.15.8 $\psi'\left(z\right)\sim\frac{1}{z}+\frac{1}{2z^{2}}+\sum_{k=1}^{\infty}\frac{B% _{2k}}{z^{2k+1}},$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\sim$: Poincaré asymptotic expansion, $\psi\left(\NVar{z}\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable A&S Ref: 6.4.12 Referenced by: §5.15 Permalink: http://dlmf.nist.gov/5.15.E8 Encodings: TeX, pMML, png See also: Annotations for §5.15 and Ch.5
 5.15.9 ${\psi}^{(n)}\left(z\right)\sim(-1)^{n-1}\left(\frac{(n-1)!}{z^{n}}+\frac{n!}{2% z^{n+1}}+\sum_{k=1}^{\infty}\frac{(2k+n-1)!}{(2k)!}\frac{B_{2k}}{z^{2k+n}}% \right).$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\sim$: Poincaré asymptotic expansion, $\psi\left(\NVar{z}\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $n$: nonnegative integer, $k$: nonnegative integer and $z$: complex variable A&S Ref: 6.4.11 Referenced by: §5.15, Erratum (V1.1.3) for Additions Permalink: http://dlmf.nist.gov/5.15.E9 Encodings: TeX, pMML, png Addition (effective with 1.1.3): This equation was added. Suggested 2021-09-09 by Calvin Khor See also: Annotations for §5.15 and Ch.5

For $B_{2k}$ see §24.2(i).

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