digamma and polygamma function

The digamma functionMathworldPlanetmath is defined as the logarithmic derivativeMathworldPlanetmath of the gamma functionDlmfDlmfMathworldPlanetmath:


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


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

ψ(z+1) =ψ(z)+1z
ψ(1-z) =ψ(z)+πcotπz
ψ(2z) =12ψ(z)+12ψ(z+12)+log2
ψ(n)(z+1) =ψ(n)(z)+(-1)nn!zn+1

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

ψ(z)=-γ-1z+k=1(1k-1z+k), (1)
ψ(n)(z)=(-1)nn!k=01(z+k)n (2)

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

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