Uniform asymptotics for jacobi polynomials with varying large negative parameters - A Riemann-Hilbert approach

An asymptotic expansion is derived for the Jacobi polynomials P _n^(α,β)(z) with varying parameters α_n = -nA + a and β_n = -nB + b, where A > 1, B > 1 and a, b are constants. Our expansion is uniformly valid in the upper half-plane ℂ̄⁺ = {z : Im z ≥ 0}. A corresponding expansion is also given for the lower half-plane ℂ̄^- = {z : Im z ≤ 0}. Our approach is based on the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou (1993). The two asymptotic expansions hold, in particular, in regions containing the curve L, which is the support of the equilibrium measure associated with these polynomials. Furthermore, it is shown that the zeros of these polynomials all lie on one side of L, and tend to L as n → ∞. © 2006 American Mathematical Society.