$m$-step rational extensions of the trigonometric Darboux-P\"{o}schl-Teller potential based on para-Jacobi polynomials

Abstract

A previous construction of regular rational extensions of the trigonometric Darboux-P\"oschl-Teller potential, obtained by one-step Darboux transformations using seed functions associated with the para-Jacobi polynomials of Calogero and Yi, is generalized by considering $m$-step Darboux transformations. As a result, some novel families of exceptional orthogonal polynomials depending on $m$ discrete parameters, as well as $m$ continuous real ones $\lambda_1$, $\lambda_2$, \ldots, $\lambda_m$, are obtained. The restrictions imposed on these parameters by the rational extensions regularity conditions are studied in detail.