  Recently, Feh\'er and Kluck discovered, at the level of classical mechanics,
new compactified trigonometric Ruijsenaars-Schneider $n$-particle systems, with
phase space symplectomorphic to the $(n-1)$-dimensional complex projective
space. In this article, we quantize the so-called type (i) instances of these
systems and explicitly solve the joint eigenvalue problem for the corresponding
quantum Hamiltonians by generalising previous results of van Diejen and Vinet.
Specifically, the quantum Hamiltonians are realized as discrete difference
operators acting in a finite-dimensional Hilbert space of complex-valued
functions supported on a uniform lattice over the classical configuration
space, and their joint eigenfunctions are constructed in terms of discretized
$A_{n-1}$ Macdonald polynomials with unitary parameters.
