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In the two preceding parts of this series of papers, we introduced andstudied a recursion scheme for constructing joint eigenfunctions $J_N(a_+,a_-,b;x,y)$ of the Hamiltonians arising in the integrable $N$-particle systemsof hyperbolic relativistic Calogero-Moser type. We focused on the first stepsof the scheme in Part I, and on the cases $N=2$ and $N=3$ in Part II. In thispaper, we determine the dominant asymptotics of a similarity transformedfunction $\rE_N(b;x,y)$ for $y_j-y_{j+1}\to\infty$, $j=1,\ldots, N-1$, andthereby confirm the long standing conjecture that the particles in thehyperbolic relativistic Calogero-Moser system exhibit soliton scattering. Thisresult generalizes a main result in Part II to all particle numbers $N>3$.

Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. III. Factorized asymptotics