  We employ a discrete integral-reflection representation of the double affine
Hecke algebra of type $C^\vee C$ at the critical level q=1, to endow the open
finite $q$-boson system with integrable boundary interactions at the lattice
ends. It is shown that the Bethe Ansatz entails a complete basis of
eigenfunctions for the commuting quantum integrals in terms of Macdonald's
three-parameter hyperoctahedral Hall-Littlewood polynomials.
