Strong Asymptotics of Jacobi-Type Kissing Polynomials

Abstract

We investigate asymptotic behavior of polynomials $p_n^{\omega }(z)$ satisfying varying non-Hermitian orthogonality relations $${\int }_{-1}^1{x}^k{p}_n^{\omega }(x)h(x)e^{\mathrm{i}\omega x}\mathrm{d}x=0,k\in \{0,\dots ,n-1\},$$ where $h(x)=h^{\ast }(x)(1-x{)}^{\alpha }(1+x{)}^{\beta },\omega =\lambda n,\lambda \ge 0$ and $h(x)$ is holomorphic and non-vanishing in a certain neighborhood in the plane. These polynomials are an extension of so-called kissing polynomials ($\alpha =\beta =0$) introduced in connection with complex Gaussian quadrature rules with uniform good properties in $\omega$.

References in corpus (2)

• Topological Expansion in the Complex Cubic Log-Gas Model. One-Cut Case
• A Gaussian quadrature rule for oscillatory integrals on a bounded interval