Perturbations around the zeros of classical orthogonal polynomials

TL;DR

This paper develops a universal linear matrix framework to analyze perturbations around the zeros of classical orthogonal polynomials, revealing eigenvalue invariance and new polynomial representations.

Contribution

It introduces a novel matrix-based approach to study zero perturbations of all classical orthogonal polynomials, including q-analogues, with unique Diophantine properties.

Findings

Eigenvalues are independent of zeros.

Eigenvectors provide polynomial representations.

Matrix exhibits remarkable Diophantine properties.

Abstract

Starting from degree N solutions of a time dependent Schroedinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the polynomial is derived. The matrix has remarkable Diophantine properties. Its eigenvalues are independent of the zeros. The corresponding eigenvectors provide the representations of the lower degree (0,1,...,N-1) polynomials in terms of the zeros of the degree N polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.