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 mu. Let the recurrence coefficients alpha(n) and beta(n) converge to 0 and 1/2, respectively. Then, q(n+1)(z)/q(n)(z) --> PHI(z), for n --> infinity, locally uniformly for z is-an-element-of C\supp mu, where PHI maps C\[-1, 1] 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 Pade approximants.