Quark Fragmentation into ^3P_J Quarkonium

We calculate the functions of parton fragmentation into $^3P_J$ quarkonium at order $\alpha_s ^2$, where the parton can be a heavy or light quark. The obtained functions explicitly satisfy the Altarelli-Parisi equation and they are divergent, behaving as $z^{-1}$ near $z=0$. However, if one choses the renormalization scale as twice of the heavy quark mass, the fragmentation functions are regular over the whole range of $z$.