X1-Jacobi Differential Polynomial Systems and Related Double-Step Shape-Invariant Liouville Potentials Solvable by Exceptional Orthogonal Polynomials

The paper develops the new formalism to treat both infinite and finite exceptional orthogonal polynomial (EOP) sequences as X-orthogonal subsets of X-Jacobi differential polynomial systems (DPSs). The new rational canonical Sturm-Liouville equations (RCSLEs) with quasi-rational solutions (q-RSs) were obtained by applying rational Rudjak-Zakhariev transformations (RRZTs) to the Jacobi equation re-written in the canonical form. The presented analysis was focused on the RRZTs leading to the canonical form of the Heun equation. It was demonstrated that the latter equation preserves its form under the second-order Darboux-Crum transformation. The associated Sturm-Liouville problems (SLPs) were formulated for the so-called ‘prime’ SLEs solved under the Dirichlet boundary conditions (DBCs). It was proven that one of the two X1-Jacobi DPSs composed of Heun polynomials contains both X1-Jacobi orthogonal polynomial system (OPS) and finite EOP sequence composed of the pseudo-Wronskian transforms of Romanovski-Jacobi (R-Jacobi) polynomials, while the second analytically-solvable Heun equation does not have the discrete energy spectrum. The quantum-mechanical realizations of the developed formalism were obtained by applying the Liouville transformation to each of the SLPs formulated in such a way.