Biorthogonal Polynomial System Composed of X-Jacobi Polynomials from Different Sequences

TL;DR

This paper explores rational Darboux transformations of Jacobi equations, leading to a new biorthogonal polynomial system composed of X-Jacobi polynomials with unique zero distributions and orthogonality properties.

Contribution

It introduces a novel biorthogonal polynomial system derived from Darboux transformations, extending classical Jacobi polynomials to include exceptional zeros and cross-orthogonality relations.

Findings

Polynomials obey cross-orthogonality when integrated from +1 to infinity.

Each X_m-Jacobi polynomial has exactly m exceptional zeros between -infinity and -1.

The system extends classical Jacobi polynomials with new zero and orthogonality properties.

Abstract

The paper examines rational Darboux transformations (RDTs) of the Jacobi equation written in the canonical form, with emphasis on the Sturm-Liouville problems (SLPs) solved under the Dirichlet boundary conditions (DBCs) at the ends of the infinite interval [1, inf). To be able to extend the analysis to the Darboux-Crum net of rational SL equations (SLEs) solved under the cited DBCs in terms of multi-indexed orthogonal exceptional Romanovski-Jacobi (XR-Jacobi) polynomials, we consider only seed functions which represent principal Frobenius solutions (PFSs) near one of the singular endpoints. . There are three distinct types of such solutions with no zeros inside the selected interval: two infinite sequences formed by the PFSs near the lower endpoint and one finite sequence formed by the PFSs near infinity. It is shown that use of classical Jacobi polynomials as seed functions results in the double-indexed manifold composed of orthogonal X_m-Jacobi polynomials in the reversed argument. As a result polynomials from this manifold obey the cross-orthogonality relation when integrated from +1 to inf. As a corollary we assert that each X_m-Jacobi polynomial of degree m + n has exactly m exceptional zeros between -inf and -1 as far as its indexes are restricted by the derived constraints on indexes of XR-Jacobi polynomials.