This paper addresses the model matching problem (MMP) for nonlinear singleinput single-output discrete-time systems. The approach is based on the infinitesimal system description in terms of the one-forms that is converted into polynomial system representation by interpreting the polynomial indeterminate as the forward shift operator acting on the one-forms. The polynomial description is then used to derive the generalized transfer function. The problem statement of the MMP (both for the feedforward and feedback cases) is given in terms of the generalized transfer function. In general, the feedforward solution exists under the restrictive conditions. Therefore, the subclass of nonlinear control systems is specified for which the solution is guaranteed to exist. The feedback solution exists always. The additional restrictions are specified for the existence of a proper compensator (in both cases). The results of the paper are illustrated by numerous examples, and the feedback solution is compared to the earlier results.