Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

Abstract

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials $P_n^{(α_n, β_n)}$ is studied, assuming that $$ \lim_{n\to\infty} \frac{α_n}{n}=A, \qquad \lim_{n\to\infty} \frac{β_n}{n}=B, $$ 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 $α_n$, $β_n$ or $α_n+β_n$ are geometrically close to $\Z$, part of the zeros accumulate along a different trajectory of the same quadratic differential.