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arXiv 1804.09697math.CAmath.SP
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正交多项式零点的静电解释

Electrostatic Interpretation of Zeros of Orthogonal Polynomials

Stefan Steinerberger

备注：
v2, minor changes, to appear in Proc. Amer. Math. Soc
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摘要：

研究了微分方程$ - (p(x) y')' + q(x) y' = λy,$，其中$p(x)$是一个至多2次的多项式，$q(x)$是一个至多1次的多项式.这包括经典的雅可比多项式、埃尔米特多项式、勒让德多项式、切比雪夫多项式和拉盖尔多项式。我们提供了这样的多项式的零点的一般静电解释：实数集$\left\{x_1, \dots, x_n\right\}$满足$$ p(x_i) \sum_{k = 1 \atop k \neq i}^{n}{\frac{2}{x_k - x_i}} = q(x_i) - p'(x_i) \qquad \mbox{for all}~ 1\leq i \leq n$$当且仅当它们是求解微分方程的多项式的零点。我们还得到了一个系统的常微分方程依赖于$p(x),q(x)$的解决方案收敛到零点的正交多项式在一个指数率。

英文摘要：

We study the differential equation $ - (p(x) y')' + q(x) y' = λy,$ where $p(x)$ is a polynomial of degree at most 2 and $q(x)$ is a polynomial of degree at most 1. This includes the classical Jacobi polynomials, Hermite polynomials, Legendre polynomials, Chebychev polynomials and Laguerre polynomials. We provide a general electrostatic interpretation of zeros of such polynomials: the set of real numbers $\left\{x_1, \dots, x_n\right\}$ satisfies $$ p(x_i) \sum_{k = 1 \atop k \neq i}^{n}{\frac{2}{x_k - x_i}} = q(x_i) - p'(x_i) \qquad \mbox{for all}~ 1\leq i \leq n$$ if and only if they are zeros of a polynomial solving the differential equation. We also derive a system of ODEs depending on $p(x),q(x)$ whose solutions converge to the zeros of the orthogonal polynomial at an exponential rate.

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