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

TL;DR

This paper investigates the distribution and asymptotic behavior of roots of a specific polynomial equation in the complex plane, providing approximations and discussing potential applications.

Contribution

It offers new asymptotic approximations for the roots of the polynomial as the degree increases, enhancing understanding of their distribution.

Findings

Asymptotic approximations for negative roots

Asymptotic approximations for positive roots

Asymptotic approximations for non-real roots

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 $$ z^{n+1}=(1+z)^n,\quad n \in \mathbb{N}.$$ 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.