标题： 复平面上多项式方程$$ z^{n+1}=(1+z)^n, n \in \mathbb{N} $$的根的分布和渐近近似分析 显示英文标题

标题： Analysis of the Distribution and Asymptotic Approximations of Roots of the Polynomial Equation $$ z^{n+1}=(1+z)^n, n \in \mathbb{N} $$ in the Complex Plane

摘要： 我们研究了序列$z_n $的正、负和非实复根的空间分布，该序列是$(n+1)$次多项式方程$$ z^{n+1}=(1+z)^n,\quad n \in \mathbb{N}.$$。我们建立了当$n\rightarrow \infty $时，该方程的负、正和非实复根序列的渐近近似。此外，我们讨论了当前问题的可能应用领域。 显示英文摘要

摘要： We study the spatial distribution of the positive, negative and non-real complex roots $z_n $ of the sequence the $(n+1)$th degree polynomial equation $$ z^{n+1}=(1+z)^n,\quad n \in \mathbb{N}.$$ We establish asymptotic approximations to the sequence of the negative, the positive and the non-real complex roots of the equation as $n\rightarrow \infty $. In addition, we discuses the possible areas of applications of the current problem.