  We solve the parametric generalized effective Schr\"odinger equation with a
specific choice of posi-tion-dependent mass function and Morse oscillator
potential by means of the Nikiforov-Uvarov (NU) method combined with the
Pekeris approximation scheme. All bound-state energies are found explicitly and
all corresponding radial wave functions are built analytically. We choose the
Weyl or Li and Kuhn ordering for the ambiguity parameters in our numerical work
to calculate the energy spectrum for a few and diatomic molecules with
arbitrary vibration and rotation quantum numbers and different
position-dependent mass functions. Two special cases including the constant
mass and the vibration s-wave (l =0) are also investigated.
