In this paper, we study the recurrence coefficients of a deformed Hermite polynomials orthogonal with respect to the weight w(x; t,α) := e−x2|x − t|α(A + B ⋅ 𝜃(x − t)),x ∈ (−∞,∞), where α > −1,A ≥ 0,A + B ≥ 0 and t ∈ ℝ. It is an extension of Chen and Feigin [J. Phys. A., Math. Gen. 39(2006) 12381–12393]. By using the ladder operator technique, we show that the recurrence coefficients satisfy a particular Painlevé IV equation and the logarithmic derivative of the associated Hankel determinant satisfies the Jimbo–Miwa–Okamoto σ form of the Painlevé IV. Furthermore, the asymptotics of the recurrence coefficients and the Hankel determinant are obtained at the hard-edge limit and can be expressed in terms of the solutions to the Painlevé XXXIV and the σ-form of the Painlevé II equation at the soft-edge limit, respectively. In addition, for the special case A = 0,B = 1, we obtain the asymptotics of the Hankel determinant at the hard-edge limit via semi-classical Laguerre polynomials with respect to the weight wα(x,t) = xαe−x2−2tx,x ∈ ℝ+, which reproduced the result in Charlier and Deano, [Integr. Geom. Methods Appl. 14(2018) 018 (p. 43)].

中文翻译：

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与 Painlevé 方程和 Heun 方程相关的退化高斯权重

在本文中，我们研究了变形 Hermite 多项式关于权重正交的递推系数 w(X; 吨,α) ：= e-X2|X - 吨|α(一种 + 乙 ⋅ 𝜃(X - 吨)),X ∈ (-∞,∞), 在哪里α > -1,一种 ≥ 0,一种 + 乙 ≥ 0和吨 ∈ ℝ. 它是 Chen 和 Feigin [J.物理。A.，数学。将军 39(2006) 12381–12393]。通过使用梯形算子技术，我们证明递归系数满足特定的 Painlevé IV 方程，并且相关 Hankel 行列式的对数导数满足 Jimbo-Miwa-OkamotoσPainlevé IV 的形式。此外，递归系数和 Hankel 行列式的渐近线是在硬边极限处获得的，可以用 Painlevé XXXIV 和σ-Painlevé II 方程在软边缘极限的形式，分别。另外，对于特殊情况一种 = 0,乙 = 1，我们通过半经典拉盖尔多项式关于权重获得汉克尔行列式在硬边极限处的渐近性wα(X,吨) = Xαe-X2-2吨X,X ∈ ℝ+，在 Charlier 和 Deano 中重现了结果，[积分。几何。方法应用程序。 14（2018）018（第 43 页）]。