Asymptotics of orthogonal polynomials for a weight with a jump on [-1,1]

TL;DR

This paper derives strong uniform asymptotics for orthogonal polynomials with a weight having a jump discontinuity on [-1,1], confirming a conjecture and analyzing local behavior near the jump using Riemann-Hilbert methods.

Contribution

It provides the first asymptotic expansion of orthogonal polynomial parameters for weights with a step-like discontinuity, including a proof of Magnus's conjecture.

Findings

Confirmed Magnus's conjecture on recurrence coefficients.

Analyzed local asymptotics at the jump point using confluent hypergeometric functions.

Showed zeros of polynomials deviate from clock behavior near the jump.

Abstract

We consider the orthogonal polynomials on $[-1,1]$ with respect to the weight $$ w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \Xi_{c}(x), \quad \alpha, \beta >-1, $$ where $h$ is real analytic and strictly positive on $[-1, 1]$, and $\Xi_{c}$ is a step-like function: $\Xi_{c}(x)=1$ for $x\in [-1, 0)$ and $\Xi_{c}(x)=c^2$, $c>0$, for $x\in [0, 1]$. We obtain strong uniform asymptotics of the monic orthogonal polynomials in $\mathbb{C}$, as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as $n\to \infty$. In particular, we prove for $w_c$ a conjecture of A. Magnus regarding the asymptotics of the recurrence coefficients. The main focus is on the local analysis at the origin. We study the asymptotics of the Christoffel-Darboux kernel in a neighborhood of the jump and show that the zeros of the orthogonal polynomials no longer exhibit the clock behavior. For the asymptotic analysis we use the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann-Hilbert problems characterizing the orthogonal polynomials. The local analysis at $x=0$ is carried out in terms of the confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest.