Strong asymptotics for jacobi polynomials with varying nonstandard parameters

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials P_n^(αn,βn) are studied, assuming that equation present 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 α_n, β_n or α_n + β_n are geometrically close to ℤ, part of the zeros accumulate along a different trajectory of the same quadratic differential.