Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomialsP _n ^α _n ^β _n are studied, assuming that

(1)

withA andB satisfyingA>−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.