, Volume 94, Issue 1, pp 195-234

Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

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Abstract

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomialsP n α n β n are studied, assuming that 1 $$\mathop {\lim }\limits_{n \to \infty } \frac{{\alpha _n }}{n} = A, \mathop {\lim }\limits_{n \to \infty } \frac{{\beta _n }}{n} = B,$$ 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 n are geometrically close to ℤ, part of the zeros accumulate along a different trajectory of the same quadratic differential.