Strong Asymptotics of Jacobi-Type Kissing Polynomials

TL;DR

This paper analyzes the asymptotic behavior of Jacobi-type kissing polynomials, extending classical results to polynomials with non-Hermitian orthogonality relations involving exponential weights and varying parameters.

Contribution

It provides new asymptotic formulas for Jacobi-type kissing polynomials with non-Hermitian orthogonality and varying weights, generalizing previous work on classical kissing polynomials.

Findings

Derived strong asymptotics for Jacobi-type kissing polynomials.

Extended analysis to non-Hermitian orthogonality with exponential weights.

Connected asymptotic behavior to complex Gaussian quadrature rules.

Abstract

We investigate asymptotic behavior of polynomials $p^{\omega}_n(z)$ satisfying varying non-Hermitian orthogonality relations $$ \int_{-1}^{1} x^kp^{\omega}_n(x)h(x) e^{\mathrm{i} \omega x}\mathrm{d} x =0, \quad k\in\{0,\ldots,n-1\}, $$ where $h(x) = h^*(x) (1 - x)^{\alpha} (1 + x)^{\beta}, \ \omega = \lambda n, \ \lambda \geq 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 $\omega$.