§ 5.15. Polygamma Functions

Permalink:: http://dlmf.nist.gov/5.15

The functions $\psi^{{(n)}}\!\left(z\right)$ , $n=1,2,\dots$ , 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) $n,m=1,2,3,\dots$ , and for $\zeta\!\left(n+1\right)$ see §Ch.25.

5.15.1 ${{\psi}^{{\prime}}}\!\left(z\right)=\sum _{{k=0}}^{\infty}\frac{1}{(k+z)^{2}},$ $z\neq 0,-1,-2,\dots$ ,

Symbols:: $\psi\!\left(z\right)$ : Psi or digamma function, $k$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.15.E1
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5.15.2 $\psi^{{(n)}}\!\left(1\right)=(-1)^{{n+1}}n!\zeta\!\left(n+1\right),$

Defines:: $\psi^{{(n)}}\!\left(z\right)$ : polygamma functions
Symbols:: $n$ : nonnegative integer
A&S Ref:: 6.4.2
Referenced by:: §5.15
Permalink:: http://dlmf.nist.gov/5.15.E2
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5.15.3 $\psi^{{(n)}}\!\left(\tfrac{1}{2}\right)=(-1)^{{n+1}}n!(2^{{n+1}}-1)\zeta\!\left(n+1\right),$

Defines:: $\psi^{{(n)}}\!\left(z\right)$ : polygamma functions
Symbols:: $n$ : nonnegative integer
A&S Ref:: 6.4.4
Permalink:: http://dlmf.nist.gov/5.15.E3
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5.15.4 ${{\psi}^{{\prime}}}\!\left(n-\tfrac{1}{2}\right)=\tfrac{1}{2}\pi^{2}-4\sum _{{k=1}}^{{n-1}}\frac{1}{(2k-1)^{2}},$

Symbols:: $\psi\!\left(z\right)$ : 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
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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, $n$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.4.6
Permalink:: http://dlmf.nist.gov/5.15.E5
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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:: $\psi\!\left(z\right)$ : 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
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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
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As $z\to\infty$ in $|\mathrm{ph}z|\le\pi-\delta\;(<\pi)$

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

Symbols:: $\psi\!\left(z\right)$ : Psi or digamma function, $\sim$ : asymptotically equal, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.4.12
Permalink:: http://dlmf.nist.gov/5.15.E8
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For $B_{{2k}}$ see §Ch.24.

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