The trigonometric Rosen-Morse potential in the supersymmetric quantum mechanics and its exact solutions

The analytic solutions of the one-dimensional Schroedinger equation for the trigonometric Rosen-Morse potential reported in the literature rely upon the Jacobi polynomials with complex indices and complex arguments. We first draw attention to the fact that the complex Jacobi polynomials have non-trivial orthogonality properties which make them uncomfortable for physics applications. Instead we here solve the above equation in terms of real orthogonal polynomials. The new solutions are used in the construction of the quantum-mechanical superpotential