  Starting from the hyperoctahedral multivariate hypergeometric function of
Heckman and Opdam (associated with the $BC_n$ root system), we arrive -- via
partial confluent limits in the sense of Oshima and Shimeno -- at solutions of
the eigenvalue equations for the Toda chain with one-sided boundary
perturbations of P\"oschl-Teller type and for the hyperbolic
Calogero-Sutherland system in a Morse potential. With the aid of corresponding
degenerations of the (bispectral dual) difference equations for the
Heckman-Opdam hyperoctahedral hypergeometric function, it is deduced that the
eigensolutions in question are holomorphic in the spectral variable.
