Interlacing properties of zeros of associated polynomials

Let w be a nonnegative integrable weight function on [−1,1] and let pn + 1(x) = xn+1 + … be the polynomial of degree n + 1 orthogonal with respect to w. Furthermore, let pn(1)(x) = xn + … denote the polynomials associated with pn + 1 and pn(1−x2)(x) = xn + … the polynomials orthogonal with respect to the weight function (1 − x2)w(x). In this paper we give necessary and sufficient conditions such that the zeros of pn(1) and pn(1−x2) strictly interlace on [−1, 1] for large n. In particular this problem is studied for the Jacobi weights wα,β(x) = (1 − x)α(1 + x)β, α,β ∈ ( −1, ∞). In this case pn(1−x2) = p′n + 1(n + 1). 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.