New Properties of the Zeros of Krall Polynomials

DOI

10.1080/14029251.2017.1375687How to use a DOI?

Keywords

Zeros of orthogonal polynomials; nonclassical orthogonal polynomials; Krall polynomials; spectral methods; pseudospectral methods

Abstract

We identify a new class of algebraic relations satisfied by the zeros of orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two, known as Krall polynomials. Given an orthogonal polynomial family {pv(x)}v=0∞ , 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 finding exact expressions for 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 identities satisfied by the Krall-Legendre, the Krall-Laguerre and the Krall-Jacobi orthogonal polynomials.

Copyright

© 2017 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC license.

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TY  - JOUR
AU  - Oksana Bihun
PY  - 2021
DA  - 2021/01/06
TI  - New Properties of the Zeros of Krall Polynomials
JO  - Journal of Nonlinear Mathematical Physics
SP  - 495
EP  - 515
VL  - 24
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2017.1375687
DO  - 10.1080/14029251.2017.1375687
ID  - Bihun2021
ER  -