Linear differential operators with polynomial coefficients generating generalised Sylvester-Kac matrices

TL;DR

This paper introduces a method to generate differential operators that solve the spectral problem for generalized Sylvester-Kac matrices, revealing new polynomial eigenfunctions and analyzing related matrix spectral properties.

Contribution

It presents a novel approach to constructing differential operators with polynomial coefficients that extend previous work by Kac, and explores spectral properties of related matrices.

Findings

Derived a first-order differential operator with polynomial eigenfunctions

Solved the spectral problem for a generalized Sylvester-Kac matrix

Analyzed spectral properties of related tridiagonal matrices

Abstract

A method of generating differential operators is used to solve the spectral problem for a generalisation of the Sylvester-Kac matrix. As a by-product, we find a linear differential operator with polynomial coefficients of the first order that has a finite sequence of polynomial eigenfunctions generalising the operator considered by M. Kac. In addition, we explain spectral properties of two related tridiagonal matrices whose shape differ from our generalisation.