§5.15 Polygamma Functions

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).

5.15.1		$${\psi }^{\prime }\left(z\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{{(k+z)}^2},$$
$z\ne 0,-1,-2,\mathrm{\dots }$,
ⓘ Symbols: $\psi \left(z\right)$: 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: $\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), ${\psi }^{(n)}\left(z\right)$: 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		$${\psi }^{(n)}\left(\frac{1}{2}\right)={(-1)}^{n+1}n!(2^{n+1}-1)\zeta \left(n+1\right),$$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), ${\psi }^{(n)}\left(z\right)$: 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		$${\psi }^{\prime }\left(n-\frac{1}{2}\right)=\frac{1}{2}{\pi }^2-4\sum _{k=1}^{n-1}\frac{1}{{(2k-1)}^2},$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: 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: $\psi \left(z\right)$: 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		$${\psi }^{(n)}\left(1-z\right)+{(-1)}^{n-1}{\psi }^{(n)}\left(z\right)={(-1)}^n\pi \frac{d^n}{{dz}^n}\cot \left(\pi z\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\psi \left(z\right)$: psi (or digamma) function, $n$: nonnegative integer and $z$: complex variable A&S Ref: 6.4.7 Permalink: http://dlmf.nist.gov/5.15.E6 Encodings: TeX, pMML, png See also: Annotations for §5.15 and Ch.5

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: $\psi \left(z\right)$: 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 $z\to \mathrm{\infty }$ in

5.15.8		$${\psi }^{\prime }\left(z\right)\sim \frac{1}{z}+\frac{1}{2z^2}+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{z^{2k+1}}.$$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\sim$: Poincaré asymptotic expansion, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable A&S Ref: 6.4.12 Permalink: http://dlmf.nist.gov/5.15.E8 Encodings: TeX, pMML, png See also: Annotations for §5.15 and Ch.5

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).