Strong Asymptotics of Jacobi-Type Kissing Polynomials

Abstract

We investigate asymptotic behaviour of polynomials pnω(z) satisfying varying non-Hermitian orthogonality relations ∫−11xkpnω(x)h(x)eiωxdx=0,k∈0,…,n−1, where h(x)=h∗(x)(1−x)α(1+x)β, ω=λn, λ≥ 0 and h(x) is holomorphic and non-vanishing in a certain neighbourhood in the plane. These polynomials are an extension of so-called kissing polynomials ( α=β=0) introduced in Asheim et al. [A Gaussian quadrature rule for oscillatory integrals on a bounded interval. Preprint, 2012 Dec 6. arXiv:1212.1293] in connection with complex Gaussian quadrature rules with uniform good properties in ω. The analysis carried out here is an extension of what was done in Celsus and Silva [Supercritical regime for the kissing polynomials. J Approx Theory. 2020 Mar 18;225:Article ID: 105408]; Deaño [Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval. J Approx Theory. 2014 Oct 1;186:33–63], and depends heavily on those works.