In a previous publication, two of us derived a relation between the scattering amplitude of three identical bosons, M_3, and a real function referred to as the divergence-free K matrix and denoted K_df,3. The result arose in the context of a relation between finite-volume energies and K_df,3, derived to all orders in the perturbative expansion of a generic low-energy effective field theory. In this work we set aside the role of the finite volume and focus on the infinite-volume relation between

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... K_df,3 and M_3. We show that, for any real choice of K_df,3, M_3 satisfies the three-particle unitarity constraint to all orders. Given that K_df,3 is also free of a class of kinematic divergences, the function may provide a useful tool for parametrizing three-body scattering data. Applications include the phenomenological analysis of experimental data (where the connection to the finite volume is irrelevant) as well as calculations in lattice quantum chromodynamics (where the volume plays a key role).