A shortcut to asymptotics for orthogonal polynomials

We consider the asymptotic behavior of the ratios q_n+1(z)/q_n(z) of polynomials orthonormal with respect to some positive measure μ. Let the recurrence coefficients _n and β_n converge to 0 and , respectively. Then, q_n+1(z)/q_n(z) Φ(z),for n→∞ locally uniformly for , where Φ maps conformally onto the exterior of the unit disc (Nevai (1979)). We provide a new and direct proof for this and some related results due to Nevai, and apply it to convergence acceleration of diagonal Padé approximants.