Interlacing properties of zeros of associated polynomials

Abstract

Let w be a nonnegative integrable weight function on [−1,1] and let p n + 1 ( x) = x n+1 + … be the polynomial of degree n + 1 orthogonal with respect to w. Furthermore, let p n (1)( x) = x n + … denote the polynomials associated with p n + 1 and p n (1− x 2) ( x) = x n + … the polynomials orthogonal with respect to the weight function (1 − x 2) w( x). In this paper we give necessary and sufficient conditions such that the zeros of p n (1) and p n (1− x 2) strictly interlace on [−1, 1] for large n. In particular this problem is studied for the Jacobi weights w α, β ( x) = (1 − x) α (1 + x) β , α, β ∈ ( −1, ∞). In this case p n (1−x 2) = p′ n + 1 (n + 1) . For a large class of parameters, including, e.g. the ultraspherical case α = β, it is shown that the interlacing property holds for each n ∈ N . Also a fairly complete description of the parameters for which the interlacing property does not hold is given.