digamma and polygamma function

The digamma function is defined as the logarithmic derivative of the gamma function:

$$\psi (z)=\frac{d}{dz}\log \mathrm{\Gamma }(z)=\frac{{\mathrm{\Gamma }}^{\prime }(z)}{\mathrm{\Gamma }(z)}.$$

Likewise the polygamma functions are defined as higher order logarithmic derivatives of the gamma function:

$${\psi }^{(n)}(z)=\frac{d^n}{d{z}^n}\log \mathrm{\Gamma }(z).$$

These equations enjoy functional equations which are closely related to those of the gamma function:

$\psi (z+1)$ $=\psi (z)+{\displaystyle \frac{1}{z}}$

$\psi (1-z)$ $=\psi (z)+\pi \cot \pi z$

$\psi (2z)$ $={\displaystyle \frac{1}{2}}\psi (z)+{\displaystyle \frac{1}{2}}\psi \left(z+{\displaystyle \frac{1}{2}}\right)+\log 2$

${\psi }^{(n)}(z+1)$ $={\psi }^{(n)}(z)+{(-1)}^n{\displaystyle \frac{n!}{z^{n+1}}}$

These functions have poles at the negative integers and can be expressed as partial fraction series:

$\psi (z)=-\gamma -{\displaystyle \frac{1}{z}}+{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}\left({\displaystyle \frac{1}{k}}-{\displaystyle \frac{1}{z+k}}\right),$ (1)

${\psi }^{(n)}(z)={(-1)}^{n}n!{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{1}{{(z+k)}^n}}$ (2)

Here, $\gamma$ is Euler–Mascheroni constant (http://planetmath.org/EulerMascheroniConstant).  Substituting  $z=1$  to (1), one gets the value

$${\mathrm{\Gamma }}^{\prime }(1)=-\gamma .$$

Title	digamma and polygamma function
Canonical name	DigammaAndPolygammaFunction
Date of creation	2013-03-22 15:53:21
Last modified on	2013-03-22 15:53:21
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	13
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 30D30
Classification	msc 33B15
Defines	digamma function
Defines	polygamma function