  Keeping in view the ordering ambiguity that arises due to the presence of
position-dependent effective mass in the kinetic energy term of the
Hamiltonian, a general scheme for obtaining algebraic solutions of quantum
mechanical systems with position-dependent effective mass is discussed. We
quantize the Hamiltonian of the pertaining system by using symmetric ordering
of the operators concerning momentum and the spatially varying mass, initially
proposed by von Roos and Levy-Leblond. The algebraic method, used to obtain the
solutions, is based on the concepts of supersymmetric quantum mechanics and
shape invariance. In order to exemplify the general formalism a class of
non-linear oscillators has been considered. This class includes the particular
example of a one-dimensional oscillator with different position-dependent
effective mass profiles. Explicit expressions for the eigenenergies and
eigenfunctions in terms of generalized Hermite polynomials are presented.
Moreover, properties of these modified Hermite polynomials, like existence of
generating function and recurrence relations among the polynomials have also
been studied. Furthermore, it has been shown that in the harmonic limit, all
the results for the linear harmonic oscillator are recovered.
