$O(\alpha_s^2)$ Contributions to the asymmetric fragmentation function in $e^+e^-$ annihilation

The order \alpha_s^2 contributions to the coefficient functions corresponding to the asymmetric fragmentation function $F_A(x,Q^2)$ in $e^+e^-$ annihilation are calculated. From this calculation we infer that the order $(\alpha_s/4\pi)^2$ correction to the flavour asymmetry sum rule is non vanishing and amounts to $-12\beta_0C_F\zeta(3)$. We also study the effect of the higher order QCD corrections on $F_A(x,Q^2)$ and compare them with the OPAL data. The latter put a strong constraint on the valence part of the fragmentation densities $D_q^H(x,\mu^2)$.