Asymptotics and zero behaviour of geometric polynomials

TL;DR

This paper investigates the asymptotic behavior and zero distribution of geometric polynomials, revealing their orthogonality and connections to Eulerian polynomials.

Contribution

It provides new asymptotic results, zero distribution analysis, and orthogonality properties for geometric polynomials, linking them to Eulerian polynomials.

Findings

Asymptotic behavior characterized in complex plane and interval (-1,0)

Distances between consecutive zeros in the bulk of (-1,0) determined

Orthogonality properties established for geometric polynomials

Abstract

We obtain some results on the asymptotic behaviour of Geometric polynomials in both the complex plane minus $[-1,0]$ and the interval $(-1,0)$. We also find the distance of consecutive zeros of these polynomials in the bulk of the interval $(-1,0)$. We also prove that they satisfy certain orthogonality properties. Due to their relationship with Eulerian polynomials, the results for Geometric polynomials can be transferred to Eulerian polynomials verbatim.