Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

Abstract

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials $P_n^{({\alpha }_n,{\beta }_n)}$ is studied, assuming that $$\underset{n\to \infty }{\lim }\frac{{\alpha }_n}{n}=A,\underset{n\to \infty }{\lim }\frac{{\beta }_n}{n}=B,$$ 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 $\Gamma$ of a certain quadratic differential according to the equilibrium measure on $\Gamma$ in an external field. However, when either ${\alpha }_n$, ${\beta }_n$ or ${\alpha }_n+{\beta }_n$ are geometrically close to $\mathbb{Z}$, part of the zeros accumulate along a different trajectory of the same quadratic differential.

31 pages, 12 figures. Some references added. To appear in Journal D'Analyse Mathematique