  We calculate the leading order QCD fragmentation functions for the production
of $P$-wave charmed beauty mesons. Long-distance effects are factored into two
nonperturbative parameters: the derivative of the radial wavefunction at the
origin and a second parameter related to the probability for a $(\bar b c)$
heavy quark pair that is produced in a color-octet $S$-wave state to form a
color-singlet $P$-wave bound state. The four $2P$ states and those $3P$ states
which lie below the $BD$ flavor threshold eventually all decay into the $1S$
ground state $B_c$ through hadronic cascades or by emitting photons. The total
fragmentation probabilities for production of the $1S$ ground state $B_c$ from
the cascades of the $2P$ and $3P$ states are about $1.7 \times 10^{-4}$ and
$2.3 \times 10^{-4}$ respectively. Thus the direct production of the $P$-wave
states via fragmentation may account for a significant fraction of the
inclusive production rate of the $B_c$ at large transverse momentum in high
energy colliders. Our analytic results for the $P$-wave fragmentation functions
disagree with those obtained earlier in the literature.
