DLMF: GA.15 Polygamma Functions

§GA.15. Polygamma Functions

Keywords:: hexagamma function $\psi^{{(4)}}\!\left(z\right)$ , pentagamma function $\psi^{{(3)}}\!\left(z\right)$ , polygamma function, tetragamma function ${{\psi}^{{\prime\prime}}}\!\left(z\right)$ , trigamma function ${{\psi}^{{\prime}}}\!\left(z\right)$

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 §.

In (GA.15.2) and (GA.15.3) $n=1,2,3,\dots$ ; for $\zeta\!\left(n+1\right)$ see Chapter ZE.

GA.15.1 ${{\psi}^{{\prime}}}\!\left(z\right)=\sum _{{k=0}}^{\infty}\frac{1}{(k+z)^{2}},$ $z\neq 0,-1,-2,\dots$ ,

Notations:: $\psi\!\left(z\right)$ : Psi or digamma function, $k$ : nonnegative integer and $z$ : complex variable
Encodings:: pMathML, png, TeX

GA.15.2 $\psi^{{(n)}}\!\left(1\right)=(-1)^{{n+1}}n!\zeta\!\left(n+1\right),$

Notations:: $\psi^{{(n)}}\!\left(z\right)$ : polygamma functions and $n$ : nonnegative integer
A&S Ref:: 6.4.2
Referenced by:: §GA.15
Encodings:: pMathML, png, TeX

GA.15.3 $\psi^{{(n)}}\!\left(\tfrac{1}{2}\right)=(-1)^{{n+1}}n!(2^{{n+1}}-1)\zeta\!\left(n+1\right),$

Notations:: $\psi^{{(n)}}\!\left(z\right)$ : polygamma functions and $n$ : nonnegative integer
A&S Ref:: 6.4.4
Referenced by:: §GA.15
Encodings:: pMathML, png, TeX

GA.15.4 ${{\psi}^{{\prime}}}\!\left(n+\tfrac{1}{2}\right)=\tfrac{1}{2}\pi^{2}-4\sum _{{k=1}}^{n}\frac{1}{(2k-1)^{2}}.$

Notations:: $\psi\!\left(z\right)$ : Psi or digamma function, $n$ : nonnegative integer and $k$ : nonnegative integer
A&S Ref:: 6.4.5
Encodings:: pMathML, png, TeX

As $z\to\infty$ in $|\mathrm{ph}z|\le\pi-\delta\;(<\pi)$

GA.15.5 ${{\psi}^{{\prime}}}\!\left(z\right)\sim\frac{1}{z}+\frac{1}{2z^{2}}+\sum _{{k=1}}^{\infty}\frac{B_{{2k}}}{z^{{2k+1}}}.$

Notations:: $\psi\!\left(z\right)$ : Psi or digamma function, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.4.12
Encodings:: pMathML, png, TeX

For $B_{{2k}}$ see §.