Angular part of the Schrödinger equation for the Hautot potential as a harmonic oscillator with a coordinate-dependent mass in a uniform gravitational field

We construct an exactly solvable model of a linear harmonic oscillator with a coordinate-dependent mass in a uniform gravitational field. This model is placed in an infinitely deep potential well with the width $2a$ and corresponds to the exact solution of the angular part of the Schrödinger equation with one of the Hautot potentials. The wave functions of the oscillator model are expressed in terms of Jacobi polynomials. In the limit $a\to\infty$, the equation of motion, wave functions, and energy spectrum of the model correctly reduce to the corresponding results of the ordinary nonrelativistic harmonic oscillator with a constant mass. We obtain a new asymptotic relation between the Jacobi and Hermite polynomials and prove it by two different methods.