We investigate asymptotic behaviour of polynomials $p_n^{\omega }\left(z\right)$ satisfying varying non-Hermitian orthogonality relations ${\int }_{-1}^1{x}^k{p}_n^{\omega }\left(x\right)h\left(x\right){\mathrm{e}}^{\mathrm{i}\omega x}\mathrm{d}x=0,k\in \{0,\dots ,n-1\},$where $h\left(x\right)=h^{\ast }\left(x\right)(1-x{)}^{\alpha }(1+x{)}^{\text{β}},\omega =\lambda n,\lambda \ge 0$ and $h\left(x\right)$ is holomorphic and non-vanishing in a certain neighbourhood in the plane. These polynomials are an extension of so-called kissing polynomials ($\alpha =\text{β}=0$) 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.

我们研究多项式的渐近行为 ${磷}_n^{\omega }\left(z\right)$ 满足不同的非厄米正交关系 ${\int }_{-1}^1{X}^{克}{磷}_n^{\omega }\left(X\right)H\left(X\right){\mathrm{电子}}^{\mathrm{一世}\omega X}\mathrm{d}X=0,克\in \{0,\dots ,n-1\},$在哪里 $H\left(X\right)=H^{＊}\left(X\right)(1-X{)}^{\alpha }(1+X{)}^{\text{β}},\omega =\lambda n,\lambda \ge 0$ 和 $H\left(X\right)$在平面的某个邻域内是全纯的且不消失。这些多项式是所谓亲吻多项式的扩展（$\alpha =\text{β}=0$) 在 Asheim 等人中介绍。[有界区间振荡积分的高斯求积法则。预印本，2012 年 12 月 6 日。arXiv:1212.1293] 与在ω 中具有均匀良好属性的复杂高斯求积规则有关。这里进行的分析是对 Celsus 和 Silva [亲吻多项式的超临界状态。J 近似理论。2020 年 3 月 18 日；225：文章 ID：105408]；Deaño [关于有界区间上的振荡权重的正交多项式的大次渐近。J 近似理论。2014 Oct 1;186:33–63]，并且在很大程度上依赖于这些作品。