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