  In order to describe the hadronization of polarized quarks, we discuss an
extension of the quark-jet model to transverse momentum dependent fragmentation
functions. The description is based on a product ansatz, where each factor in
the product represents one of the transverse momentum dependent splitting
functions, which can be calculated by using effective quark theories. The
resulting integral equations and sum rules are discussed in detail for the case
of inclusive pion production. In particular, we demonstrate that the
3-dimensional momentum sum rules are satisfied naturally in this transverse
momentum dependent quark-jet model. Our results are well suited for numerical
calculations in effective quark theories, and can be implemented in Monte-Carlo
simulations of polarized quark hadronization processes.
