On Finite Exceptional Orthogonal Polynomial Sequences Composed of Rational Darboux Transforms of Romanovski-Jacobi Polynomials

The paper presents the united analysis of the finite exceptional orthogonal polynomial (EOP) sequences composed of rational Darboux transforms of Romanovski-Jacobi polynomials. It is shown that there are four distinguished exceptional differential polynomial systems (X-Jacobi DPSs) of series J1, J2, J3, and W. The first three X-DPSs formed by pseudo-Wronskians of two Jacobi polynomials contain both exceptional orthogonal polynomial systems (X-Jacobi OPSs) on the interval (-1,+1) and the finite EOP sequences on the positive interval (1,Inf). On the contrary, the X-DPS of series W formed by Wronskians of two Jacobi polynomials contains only (infinitely many) finite EOP sequences on the interval (1,Inf). In addition, the paper rigorously examines the three isospectral families of the associated Liouville potentials (rationally extended hyperbolic Pöschl-Teller potentials of types a, b, and a’) exactly quantized by the EOPs in question.