  Generalized Z_k^(r/2) parafermionic theories - characterized by the dimension
(r/2)(1-1/k) of the basic parafermionic field - provide potentially interesting
quantum-Hall trial wavefunctions. Such wavefunctions reveal a W_k structure.
This suggests the equivalence of (a subclass of) the Z_k^(r/2) models and the
W_k(k+1,k+r) ones. This is demonstrated here for r=3 (the gaffnian series). The
agreement of the parafermionic and the W spectra relies on the prior
determination of the field identifications in the parafermionic case.
