The phonon modes of a quantum well wire, formed by a cylindrical polar semiconductor 1 (well material) (ρ<R₁) embedded in another polar semiconductor 2 (barrier material) (R₁≤ρ<R₂), were studied by using the dielectric continuum model. The confined longitudinal optical phonon modes both in the wire (LO1) and in the barrier materials (LO2) and the interface optical (IO) phonon modes as well as the corresponding electron-phonon interaction Hamiltonians are derived. With a two-parameter variational trial wave function, the bound-polaron binding energy has been calculated numerically for different confining potential heights and wire radii. The result shows that the electron-phonon interaction can greatly modify the impurity binding energy. The IO modes are the main factor contributing to the modification. The influence of the LO1 modes increases as the wire radius increases and reaches the bulk limit at large wire radius, while the LO2 modes only show their influence at narrow wire radius.