5 Gamma Function Properties 5.14 Multidimensional Integrals 5.16 Sums

§5.15 Polygamma Functions

ⓘ

Defines:: $\psi^{(\NVar{n})}\left(\NVar{z}\right)$ : polygamma functions
Keywords:: continued fractions, definition, polygamma functions, properties, special values
Referenced by:: §5.4(ii)
Permalink:: http://dlmf.nist.gov/5.15
Addition (effective with 1.1.3):: Equation (5.15.9) was added. In the sentence immediately above (5.15.8), “ $|\operatorname{ph}z|\leq\pi-\delta\;(<\pi)$ ” was replaced with “ $|\operatorname{ph}z|\leq\pi-\delta$ ”.
See also:: Annotations for Ch.5

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		${\psi}^{(n)}\left(1-z\right)+(-1)^{n-1}{\psi}^{(n)}\left(z\right)=(-1)^{n}\pi% \frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\cot\left(\pi z\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\psi\left(\NVar{z}\right)$ : 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: $\psi\left(\NVar{z}\right)$ : 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).

5.14 Multidimensional Integrals 5.16 Sums

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§5.15 Polygamma Functions

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