对称化过程与截断正交多项式
Symmetrization process and truncated orthogonal polynomials
AI总结：
该研究定义了关于特定线性泛函正交的截断Laguerre多项式族，结合对称化得到的多项式与三项递推关系，分析其性质、系数渐近展开及相关Painlevé方程，并给出零点的静电解释与动力学特性。
摘要：
我们定义了截断Laguerre多项式族$P_n(x;z)$，其关于由$$\rangle{\rangle,p\rangle}=\rangle_{0}^zp(x)x^αe^{-x}dx,\qquadα>-1$$定义的线性泛函$\ell$正交。$P_n(x;z)$与通过对称化过程得到的多项式$S_n(x;z)$之间的关联是我们分析的核心要素。据此，我们结合两类多项式满足的三项递推关系的参数关联，研究了$P_n(x;z)$和$S_n(x;z)$的若干性质，给出了这些系数的渐近展开式，自然导出了与这些系数相关的离散Painlevé方程与Painlevé方程，还给出了这类多项式零点的静电解释以及零点随参数$z$变化的动力学特性。
英文摘要：
We define the family of truncated Laguerre polynomials $P_n(x;z)$, orthogonal with respect to the linear functional $\ell$ defined by $$\langle{\ell,p\rangle}=\int_{0}^zp(x)x^αe^{-x}dx,\qquadα>-1.$$ The connection between $P_n(x;z)$ and the polynomials $S_n(x;z)$ (obtained through the symmetrization process) constitutes a key element in our analysis. As a consequence, several properties of the polynomials $P_n(x;z)$ and $S_n(x;z)$ are studied taking into account the relation between the parameters of the three-term recurrence relations that they satisfy. Asymptotic expansions of these coefficients are given. Discrete Painlevé and Painlevé equations associated with such coefficients appear in a natural way. An electrostatic interpretation of the zeros of such polynomials as well as the dynamics of the zeros in terms of the parameter $z$ are given.
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