New Properties of the Zeros of Krall Polynomials

TL;DR

This paper discovers new algebraic relations among the zeros of Krall orthogonal polynomials, extending classical properties to nonclassical families and revealing identities involving zeros of different degrees.

Contribution

It introduces a novel framework relating zeros of Krall polynomials of different degrees via spectral and pseudospectral matrix transformations.

Findings

Derived algebraic relations for zeros of Krall polynomials.

Established identities linking zeros of different polynomial degrees.

Extended classical orthogonal polynomial properties to Krall families.

Abstract

We identify a class of remarkable algebraic relations satisfied by the zeros of the Krall orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two. Given an orthogonal polynomial family {p_n(x)}, we relate the zeros of the polynomial p_N with the zeros of p_m for each m <= N (the case m=N corresponding to the relations that involve the zeros of p_N only). These identities are obtained by exacting the similarity transformation that relates the spectral and the (interpolatory) pseudospectral matrix representations of linear differential operators, while using the zeros of the polynomial p_N as the interpolation nodes. The proposed framework generalizes known properties of classical orthogonal polynomials to the case of nonclassical polynomial families of Krall type. We illustrate the general result by proving new remarkable identities satisfied by the Krall-Legendre, the Krall-Laguerre and the Krall-Jacobi orthogonal polynomials.