标题： 多项式在重复极微分下的渐近根分布 显示英文标题

标题： Asymptotic root distribution of polynomials under repeated polar differentiation

摘要： 给定一个具有固定渐近根分布的实根多项式序列$\{p_n\}_{n\geq 1}$，我们研究该序列的重复极导数的渐近根分布。 这种极限分布可以看作是分数自由卷积和沿Möbius变换的分布前推映射的结果。 这个新的测度操作家族形成一个半群，并满足一些其他良好的性质。 利用极导数彼此交换的事实，我们得到了这些新操作之间的一个非平凡交换关系。 我们还研究了极自由无限可分性的概念，并构造了类似Belinschi-Nica的半群。 最后，我们提供了一些在这些新操作下表现良好的分布的例子，包括Marchenko-Pastur分布和柯西分布。 显示英文摘要

摘要： Given a sequence of real rooted polynomials $\{p_n\}_{n\geq 1}$ with a fixed asymptotic root distribution, we study the asymptotic root distribution of the repeated polar derivatives of this sequence. This limiting distribution can be seen as the result of fractional free convolution and pushforward maps along M\"obius transforms for distributions. This new family of operations on measures forms a semigroup and satisfy some other nice properties. Using the fact that polar derivatives commute with one another, we obtain a non-trivial commutation relation between these new operations. We also study a notion of polar free infinite divisibility and construct Belinschi-Nica type semigroups. Finally, we provide some interesting examples of distributions that behave nicely with respect to these new operations, including the Marchenko-Pastur and the Cauchy distributions.