  The production of vector mesons in the fragmentation process of polarized
quarks is studied within the recursive String+${}^3P_0$ model, improving a
previous version of the model in which the production of pseudoscalar mesons
only was considered. Two types of couplings of the vector meson to quarks are
introduced, their coupling constants being the additional free parameters of
the model. The angular distribution of the decay products of the polarized
vector meson is deduced from the spin density matrix of the meson and the spin
information is propagated along the fragmentation chain taking into account the
entanglement of spin states. The new model is implemented in a stand alone
Monte Carlo program utilized to investigate in detail kinematic distributions
and transverse spin asymmetries. The sensitivity of these observables to the
new free parameters is discussed and the Monte Carlo results are compared with
experimental data on transverse spin asymmetries.
