Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials P (αn,βn) n is studied, assuming that lim n→∞ αn n = A , lim n→∞ βn n = B , with A and B satisfying A > −1, B > −1, A + B < −1. The asymptotic analysis is based on the non-Hermitian orthogonality of these polynomials, and uses the Deift/Zhou steepest descent analysis for matrix Riemann-Hilbert problems. As a corollary, asymptotic zero behavior is derived. We show that in a generic case the zeros distribute on the set of critical trajectories Γ of a certain quadratic differential according to the equilibrium measure on Γ in an external field. However, when either αn, βn or αn + βn are geometrically close to Z, part of the zeros accumulate along a different trajectory of the same quadratic differential.