正交和多重正交多项式的静电伙伴和零点
Electrostatic partners and zeros of orthogonal and multiple orthogonal polynomials
摘要：
对于一个给定的多项式$P$与简单的零，和一个给定的半经典重量$w$，我们提出了一个建设，产生一个线性二阶微分方程（ODE），并因此，静电模型的零点$P$。这个常微分方程的系数用对偶多项式表示，我们称之为$P$的静电伙伴。这种构造是绝对一般的，并且可以对复平面上的任何具有单零点和任何半经典权的多项式进行。一个额外的假设准正交的$P$相对于$w$，使我们能够给出更精确的界限上的程度的静电伙伴。在正交和拟正交多项式的情形下，我们恢复了一些已知的结果，并推广了其它的结果。此外，对于埃尔米特-Padé或多个正交多项式的类型II，这种方法产生一个系统的线性二阶微分方程，从那里我们得到一个静电的解释，他们的零的矢量平衡。在Angelesco、Nikishin和广义Nikishin系统的特殊情况下得到了更详细的结果。我们还讨论了这些模型的离散到连续过渡的渐近制度，作为零点的数目趋于无穷大，到已知的向量平衡问题。最后，我们讨论了如何获得二阶常微分方程的系统产生一个三阶微分方程这些多项式，以及在文献中描述。最后，我们提出了几个说明性的例子。
英文摘要：
For a given polynomial $P$ with simple zeros, and a given semiclassical weight $w$, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of $P$. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of $P$. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of $P$ with respect to $w$ allows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasi-orthogonal polynomials, we recover some of the known results and generalize others. Additionally, for the Hermite--Padé or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems. We also discuss the discrete-to-continuous transition of these models in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
