$a$-potent Schwarz matrices and Bessel-like Jacobi polynomials

TL;DR

This paper investigates reconstructing Schwarz matrices from a single eigenvalue, linking the problem to a class of Jacobi orthogonal polynomials that serve as a finite analogue of Bessel polynomials.

Contribution

It introduces a novel inverse eigenvalue problem for Schwarz matrices and connects it to Bessel-like Jacobi polynomials, expanding understanding of their discrete analogues.

Findings

Reconstruction of Schwarz matrices from one eigenvalue is possible.

Identifies a connection between Schwarz matrices and Bessel-like Jacobi polynomials.

Provides a framework for finite analogues of Bessel polynomials.

Abstract

We consider the problem of the reconstruction of a Schwarz matrix from exactly one given eigenvalue. This inverse eigenvalue problem leads to the Jacobi orthogonal polynomials~$\{P_k^{(-n,n)}\}_{k=0}^{n-1}$ that can be treated as a discrete finite analogue of Bessel polynomials.