Exceptional Differential Polynomial Systems Formed by Simple Pseudo-Wronskians of Jacobi Polynomials and Their Infinite and Finite X-Orthogonal reductions

The paper advances the new technique for constructing the exceptional differential polynomial systems (X-DPSs) and their infinite and finite orthogonal subsets. First, using Wronskians of Jacobi polynomials (JPWs) with a common pair of the `indexes, we generate the Darboux-Crum nets of the rational canonical Sturm-Liouville equations (RCSLEs). It is shown that each RCSLE in question has four infinite sequences of quasi-rational solutions (q-RSs) such that the polynomial components from each sequence form a X-Jacobi DPS composed of simple pseudo-Wronskian polynomials (p-WPs). For each p-th order rational Darboux Crum transform of the Jacobi-reference (JRef) CSLE used as the starting point, we formulate two rational Sturm-Liouville problems (RSLPs) by imposing the Dirichlet boundary conditions on the solutions of the so-called ‘prime’ SLE at the ends of the intervals (-1,+1) or (+1,∞). Finally, we demonstrate that the polynomial components of the q-RSs representing the eigenfunctions of these two problems have the form of simple p-WPs composed of p Romanovski-Jacobi (R-Jacobi) polynomials with the same pair of the indexes and a single classical Jacobi polynomial or accordingly p classical Jacobi polynomials with the same pair of positive indexes and a single R-Jacobi polynomial. The common fundamentally important feature of all the simple p-WPs involved is that they do not vanish at the finite singular endpoints –the main reason of why they were selected for the current analysis in the first place. The discussion is accompanied by a sketch of the one-dimensional quantum-mechanical problems exactly solvable by the aforementioned infinite and finite EOP sequences.