Complex energies and the Lambert $W$ function

We apply the Lambert $W$ function in the context of complex energy values in statistical systems of fermions and bosons. We derive the condition for the transformation connecting fermions and bosons. We discuss the physical significance of these results and investigate the conditions under which bosonization effects take place. The fermion and boson statistical and structural distributions discussed in this work suggest the possibility of their extensions to generalized Planck distributions.