$O(\alpha_s^2)$ Timelike Wilson Coefficients for Parton-Fragmentation Functions in Mellin Space

Abstract

We calculate the Mellin moments of the $O(\alpha_s^2)$ coefficient functions for the unpolarized and polarized fragmentation functions. They can be expressed in terms of multiple finite harmonic sums of maximal weight {\sf w = 4}. Using algebraic and structural relations between harmonic sums one finds that besides the single harmonic sums only three basic sums and their derivatives w.r.t. the summation index contribute. The Mellin moments are analytically continued to complex values of the Mellin variable. This representation significantly reduces the large complexity being present in $x$--space calculations and allow very compact and fast numerical implementations.