Orthogonal Polynomials from Hermitian Matrices

TL;DR

This paper develops a unified matrix-based framework for orthogonal polynomials derived from Hermitian matrices, connecting them to exactly solvable quantum systems and explicitly deriving key properties.

Contribution

It introduces a matrix approach to orthogonal polynomials using Hermitian matrices, establishing duality and explicit formulas for polynomial properties.

Findings

Duality relation between eigenpolynomials and dual polynomials

Explicit formulas for recurrence coefficients and normalization constants

Connection to exactly solvable Schrödinger equations

Abstract

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger equations. The hermitian matrices (factorisable Hamiltonians) are real symmetric tri-diagonal (Jacobi) matrices corresponding to second order difference equations. By solving the eigenvalue problem in two different ways, the duality relation of the eigenpolynomials and their dual polynomials is explicitly established. Through the techniques of exact Heisenberg operator solution and shape invariance, various quantities, the two types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the coefficients of the three term recurrence, the normalisation measures and the normalisation constants etc. are determined explicitly.