Orthogonal polynomials from Hermitian matrices

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 Schrödinger equations. The Hermitian matrices (factorizable Hamiltonians) are real symmetric tridiagonal (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 normalization measures and the normalisation constants, etc., are determined explicitly.