%e calculate the process-independent fragmentation functions for a b antiquark to &agment into longitudinally and transversely polarized B; ( Sz) mesons to leading order in the +CD strong coupling constant. In the special case of equal quark mass, we recover previous results for the f'ragmentation of c m Q and b + T. Various spin asymmetry parameters are de6ned as measures of the relative population of the longitudinally and transversely polarized vector meson states. In the heavy quark mass limit m&~oo, our polarized fragmentation functions obey heavy quark spin symmetry, we therefore apply them as a model to describe the fragmentation of charm and bottom quarks into heavy-light mesons like D' snd B' Th.e spin asymmetry parameter, n(z), is consistent with the existing CLEO data for D' The s. caling behavior of (z) is studied in detail. We Snd excellent agreement between the predictions of (z) trom our fragmentation functions and the experimental data for D' and B' from the CERN LEP, CLEO, and ARGUS detectors. Finally, we also point out that the spin asymmetry depends signi6cantly on the transverse momentum pz of the vector mesons relative to the &agmentation axis.PACS number(s): 13. 87.Fh, 12.39. Hg, 13. 88.+e, 14.40.n lyzer of the heavy quark spin. Several recent works (Refs. [4 -8]) have been devoted to the polarization effects in the and Ap baryons at the CERN Large Electron-Positron Collider (LEP) energy. In this paper, we will investigate in detail the spin alignment of a heavy quark that undergoes &agxnentation into the spin triplet 8-wave meson state.Recently, it has been pointed out [9] that the dominant production of (bc) mesons at the large transverse momen-turn region is due to &agmentation, in which a high energy 6 antiquark is produced &om a hard process and subsequently fragments into various (bc) meson states. Furthermore, the corresponding process-independent fragmentation functions of b -+ B, (~S )sand b~B ;( Sq) were calculated within perturbative QCD (PQCD) [9,10] to leading order in both the strong coupling constant a, and v, where v is the typical velocity of the charm quark inside the meson. A lower bound of the inclusive branching &action for the production of B, at Z has been estimated to be about 2.3 x 10 4 [9], including both 1S and 2S states. It implies that only about 230 B+ or 230 B, are produced &om 10 Z . At the Fermilab Tevatron with a luminosity of 25 pb, one expects [ll] about 2 x 104 B mesons to be produced with pz (B, ) ) 10 GeV &om the direct 6 antiquark and induced gluon &agmentation. The three charged leptons &oxn a secondary vertex observed in the decay B+~J /Q+ E'vt followed by J/1b -+ l'f, where f, f' = e, p, can provide a clean signature for B,. The coxnbined branching ratio of these decays is expected to be 0.2%, which implies 40 distinct events at the Tevatron. In addition, the B meson can be fully reconstructed via hadronic decay modes, e.g. , B+~J /@ + sr+, with J/@~f /. Thus, unless LEP can increase its l~~minosity by an order of xnagnitude or so in the near future...