On two direct limits relating pseudo-Jacobi polynomials to Hermite polynomials and the pseudo-Jacobi oscillator in a homogeneous gravitational field

We present two new limit relations that reduce the orthogonal pseudo-Jacobi polynomials directly to the Hermite polynomials with shifted and nonshifted arguments. The proofs of these limit relations are based on the method of mathematical induction. These limits open up the prospects for studying new exactly solvable harmonic oscillator models in homogeneous external fields in quantum mechanics in terms of pseudo-Jacobi polynomials. As an application of these limit relations, a model of a linear harmonic oscillator with a position-dependent mass in an external homogeneous gravitational field (a pseudo-Jacobi oscillator in an external field) is considered. The form of the generalized Hamiltonian for describing quantum mechanical systems with a position-dependent mass is presented.