Asymptotic formulae for the zeros of orthogonal polynomials

Let $p_n(t)$ be an algebraic polynomial that is orthonormal with weight $p(t)$ on the interval $[-1, 1]$. When $p(t)$ is a perturbation (in certain limits) of the Chebyshev weight of the first kind, the zeros of the polynomial $p_n(\cos\tau)$ and the differences between pairs of (not necessarily consecutive) zeros are shown to satisfy asymptotic formulae as $n\to\infty$, which hold uniformly with respect to the indices of the zeros. Similar results are also obtained for perturbations of the Chebyshev weight of the second kind. First, some preliminary results on the asymptotic behaviour of the difference between two zeros of an orthogonal trigonometric polynomial, which are needed, are established.
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