We study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex-valued weight function, , over the interval , where is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, , due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as have recently been studied for , and our main goal is to extend these results to all in the complex plane. We first use the technique of continuation in parameter space, developed in the context of the theory of integrable systems, to extend previous results on the so-called modified external field from the imaginary axis to the complex plane minus a set of critical curves, called breaking curves. We then apply the powerful method of nonlinear steepest descent for oscillatory Riemann–Hilbert problems developed by Deift and Zhou in the 1990s to obtain asymptotics of the recurrence coefficients of these polynomials when the parameter is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter approaches a breaking curve, by considering double scaling limits as approaches these points. We see a qualitative difference in the behavior of the recurrence coefficients, depending on whether or not we are approaching the points or some other points on the breaking curve.

中文翻译：

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Kissing 多项式的全局相图和大阶渐近性

我们研究了一族单数正交多项式，它们在区间上与可变的复值权重函数 正交，其中是任意的。这个多项式族最初出现在文献中，当时参数是纯虚数，也就是说，由于它与高振荡积分的复杂高斯求积规则有关。最近研究了这些多项式的渐近性，我们的主要目标是将这些结果扩展到所有在复平面。我们首先使用在可积系统理论背景下开发的参数空间中的延拓技术，将先前关于所谓的修正外场的结果从虚轴扩展到减去一组临界曲线的复平面，称为打破曲线。然后，我们将 Deift 和 Zhou 在 1990 年代开发的强大的非线性最速下降方法应用于振荡黎曼-希尔伯特问题，以获得参数远离断裂曲线时这些多项式的递推系数的渐近性。然后，我们通过考虑双标度限制，提供参数接近断裂曲线时的递推系数分析接近这些点。我们看到递归系数的行为存在质的差异，这取决于我们是在接近断裂曲线上的点还是其他点。