Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Zero Distribution of Polynomials Generated by a Power of a Cubic Polynomial

Travis Steele, Khang Tran

TL;DR

This work studies the zero distribution of the polynomial sequence {P_m^(α)(z)} generated by 1/(1+t+zt^3)^α. Using an explicit coefficient form, Cauchy integral representations, and contour deformation, the authors obtain a Watson-type asymptotic analysis showing that, for α∈(0,1) and large m, all zeros lie on the real interval (-∞, -4/27) and accumulate with a precise limiting density expressed in terms of x, the real root of 1+t+zt^3. They further show that the zeros can be counted via a winding-number argument, obtaining at least ⌊m/3⌋ zeros on the curve and hence all zeros lie there; an extension to all α>0 is achieved through a derivative identity. The results extend the theory of zero distributions for polynomial sequences generated by cubic polynomials and quantify the asymptotic distribution of zeros with an explicit density function, contributing to the understanding of non-classical orthogonality and asymptotic root behavior in complex plane settings.

Abstract

For each $α>0$ and $A(z),B(z)\in\mathbb{C}[z]$, we study the zero distribution of the sequence of polynomials $\left\{ P_{m}^{(α)}(z)\right\} _{m=0}^{\infty}$ generated by $(1+B(z)t+A(z)t^{3})^{-α}$. We show that for large $m$, the zeros of $P_{m}^{(α)}(z)$ lie on an explicit curve on the complex plane.

Zero Distribution of Polynomials Generated by a Power of a Cubic Polynomial

TL;DR

This work studies the zero distribution of the polynomial sequence {P_m^(α)(z)} generated by 1/(1+t+zt^3)^α. Using an explicit coefficient form, Cauchy integral representations, and contour deformation, the authors obtain a Watson-type asymptotic analysis showing that, for α∈(0,1) and large m, all zeros lie on the real interval (-∞, -4/27) and accumulate with a precise limiting density expressed in terms of x, the real root of 1+t+zt^3. They further show that the zeros can be counted via a winding-number argument, obtaining at least ⌊m/3⌋ zeros on the curve and hence all zeros lie there; an extension to all α>0 is achieved through a derivative identity. The results extend the theory of zero distributions for polynomial sequences generated by cubic polynomials and quantify the asymptotic distribution of zeros with an explicit density function, contributing to the understanding of non-classical orthogonality and asymptotic root behavior in complex plane settings.

Abstract

For each and , we study the zero distribution of the sequence of polynomials generated by . We show that for large , the zeros of lie on an explicit curve on the complex plane.
Paper Structure (5 sections, 17 theorems, 197 equations, 5 figures)

This paper contains 5 sections, 17 theorems, 197 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Basic Properties
  3. A formula for $P_{m}^{(\alpha)}(z)$
  4. Asymptotic Behavior
  5. Main Results

Key Result

Theorem 1

For $\alpha>0$ and $A(z),B(z)\in\mathbb{C}[z]$, let $\left\{ H_{m}^{(\alpha)}(z)\right\} _{m=0}^{\infty}$ be the sequence of polynomials generated by Then for all $m$ sufficiently large, the zeros of $H_{m}^{(\alpha)}(z)$ lie on the curve defined by

Figures (5)

  • Figure 1.1: Zeros of $P_{m}^{(\alpha)}(z)$ for $1\leq m\leq50$ and $\alpha=7.5$.
  • Figure 1.2: Limiting probability density function of the zeros of $\{P_{m}^{(\alpha)}(z)\}_{m=0}^{\infty}$. The vertical dashed line is drawn at $z=-\frac{4}{27}$.
  • Figure 3.1: The contour $\gamma$
  • Figure 4.1: The contour integral of $B_{m}(t,\theta)$
  • Figure 5.1: $\operatorname{Arg}(x-y)$

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • Lemma 6
  • proof
  • Theorem 7
  • ...and 19 more