Accurate approximations to the polygamma functions

Abstract

Empirically obtained approximations to the polygamma functions ψ ( n ) ( x ) {\psi ^{\left ( n \right )}}\left ( x \right ) and their associated accuracies are presented. These simple functions are quite accurate, this accuracy increasing with x x and being best for small n n . These approximate expressions are shown to be circuitously related to the Stirling approximation to the parent gamma function.