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

TL;DR

This paper generalizes the construction of rational extensions of the trigonometric Darboux-P"oschl-Teller potential using multi-step Darboux transformations, leading to new families of exceptional orthogonal polynomials with multiple parameters.

Contribution

It introduces an $m$-step Darboux transformation framework to generate novel families of exceptional orthogonal polynomials depending on discrete and continuous parameters.

Findings

New families of exceptional orthogonal polynomials are derived.

Parameter restrictions for regularity are thoroughly analyzed.

Multi-parameter extensions broaden the class of solvable potentials.

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.