A new model of the linear harmonic oscillator with the position-dependent mass and two direct limits relating Jacobi polynomials to Hermite polynomials

We constructed a new exactly solvable model of the non-relativistic quantum linear harmonic oscillator with the position-dependent mass Mx=2m0/eαx+1. We solved the Schrödinger equation for this model in the absence and presence of an external homogeneous field. We showed that in both cases the wave functions of the bound states are expressed through Jacobi polynomials, and the corresponding energy spectra are non-equidistant. We also showed that in the limit α→0 the wave functions and energy spectra coincide with the corresponding expressions for a linear harmonic oscillator with the constant mass m0 in the absence and presence of an external homogeneous field. We present also two new limit relations that reduce the Jacobi polynomials directly to the Hermite polynomials with shifted and non-shifted arguments. The proofs of these limit relations are based on the method of mathematical induction.