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

Abstract

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

29 Pages, 4 Tables, 8 figures