Blumenthal's Theorem for Laurent Orthogonal Polynomials

Abstract

We investigate polynomials satisfying a three-term recurrence relation of the form B n ( x)=( x− β n ) B n−1 ( x)− α n x B n−2 ( x), with positive recurrence coefficients α n+1 , β n ( n=1,2,…). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. We analyse in more detail the case where α n → α and β n → β and show that the zeros of B n are dense on an interval and that the support of the Laurent orthogonality measure is equal to this interval and a set which is at most denumerable with accumulation points (if any) at the endpoints of the interval. This result is the Laurent version of Blumenthal's theorem for orthogonal polynomials.