Baker-Akhiezer specialisation of joint eigenfunctions for hyperbolic relativistic Calogero-Moser Hamiltonians

In earlier joint work with Ruijsenaars, we constructed and studied symmetric joint eigenfunctions $J_N$ for quantum Hamiltonians of the hyperbolic relativistic $N$-particle Calogero--Moser system. For generic coupling values, they are non-elementary functions that in the $N=2$ case essentially amount to a `relativistic' generalisation of the conical function specialisation of the Gauss hypergeometric function ${}_2F_1$. In this paper, we consider a discrete set of coupling values for which the solution to the joint eigenvalue problem is known to be given by functions $\psi_N$ of Baker--Akhiezer type, which are elementary, but highly nontrivial, functions. Specifically, we show that $J_N$ essentially amounts to the antisymmetrisation of $\psi_N$ and, as a byproduct, we obtain a recursive construction of $\psi_N$ in terms of an iterated residue formula.