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In a previous paper, we introduced and developed a recursive construction of joint eigenfunctions   $J_N(a_+,a_-,b;x,y)$   for the Hamiltonians of the hyperbolic relativistic Calogero–Moser system with arbitrary particle number   $N$  . In this article, we focus on the cases   $N=2$   and   $N=3$  , and establish a number of conjectured features of the corresponding joint eigenfunctions. More specifically, choosing   $a_+,a_-$   positive, we prove that   $J_2(b;x,y)$   and   $J_3(b;x,y)$   extend to globally meromorphic functions that satisfy various invariance properties as well as a duality relation. We also obtain detailed information on the asymptotic behavior of similarity transformed functions   ${\rm E}_2(b;x,y)$   and   ${\rm E}_3(b;x,y)$  . In particular, we determine the dominant asymptotics for   $y_1-y_2\to\infty$   and   $y_1-y_2,y_2-y_3\to\infty$  , respectively, from which the conjectured factorized scattering can be read off.

Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type II. The Two- and Three-Variable Cases