Strong Asymptotics of Jacobi-Type Kissing Polynomials

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^*(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 $\omega$.