The slow relaxation, the diffusivity, and the fast cage-dynamics of a melt of fully flexible unentangled polymer chains is studied by molecular-dynamics simulations. States with different nonbonding potential, chain length, density and temperature are considered. The scaling between the slow dynamics and the fast dynamics, as characterized by the amplitude of the rattling motion inside the cage, is evidenced. The analysis carried out in terms of the van Hove function shows that: (i) the scaling does not depend on the specific quantity used to quantify both the relaxation times and the amplitude of the rattling motion; (ii) it holds on the length scale of the jump-like dynamics; (iii) it also holds on the time scale of the diffusive regime if the chain-length effect is taken into proper account, thus extending analogous results already known for atomic liquids.