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Some properties of the quadrinomials $p(z)=1+κ(z+z^{N-1})+z^N$ and $q(z)=1+κ(z-z^{N-1})-z^N$

Dmitriy Dmitrishin, Alexander Stokolos

Abstract

We show that all the zeros of the quadrinomial $p(z)=1+κ(z+z^{N-1})+z^N$ lie on the unit circle if and only if the inequalities \[ -1\leκ\le 1\; (\mbox{ if $N$ is even}),\;\; -1\leκ\le N/(N-2)\; (\mbox{ if $N$ is odd}) \] hold. For the quadrinomial $q(z)=1+κ(z-z^{N-1})-z^N$, the corresponding inequalities are \[ -N/(N-2)\leκ\le 1\; (\text{ if $N$ is odd}),\;\; -N/(N-2)\leκ\le N/(N-2)\; (\text{ if $N$ is even}). \] In the cases of limiting values of the parameter $κ$, we provide factorization formulas for the corresponding quadrinomials. For example, when $N$ is odd and $κ=N/(N-2)$, the following representation is valid: \[ p(z)=(1+z)^3\prod_{j=1}^{(N-3)/2}[1+z^2-2zγ_j], \] where $γ_j=1-2ν_j^2$ with $\{ν_j\}_{j=1}^{(N-3)/2}$ being the collection of positive roots of the equation $U'_{N-2}(x)=0$; here \[ U_j(x)=U_j(\cos t)=\frac{\sin(j+1)t}{\sin t}=2^j x^j+\ldots \] are Chebyshev polynomials of the second kind and $U'_j(x)$ are their derivatives. Similar factorization formulas are also provided for $q(z)$. As an application of the obtained results, we give the factorization formulas for the derivative of the Fejér polynomial, as well as construct certain univalent polynomials related to the polynomials $p(z)$ and $q(z)$.

Some properties of the quadrinomials $p(z)=1+κ(z+z^{N-1})+z^N$ and $q(z)=1+κ(z-z^{N-1})-z^N$

Abstract

We show that all the zeros of the quadrinomial lie on the unit circle if and only if the inequalities hold. For the quadrinomial , the corresponding inequalities are In the cases of limiting values of the parameter , we provide factorization formulas for the corresponding quadrinomials. For example, when is odd and , the following representation is valid: \[ p(z)=(1+z)^3\prod_{j=1}^{(N-3)/2}[1+z^2-2zγ_j], \] where with being the collection of positive roots of the equation ; here are Chebyshev polynomials of the second kind and are their derivatives. Similar factorization formulas are also provided for . As an application of the obtained results, we give the factorization formulas for the derivative of the Fejér polynomial, as well as construct certain univalent polynomials related to the polynomials and .

Paper Structure

This paper contains 4 sections, 9 theorems, 48 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Factorization of the derivative of the Fejér polynomial
  4. Quadrinomial $p(z)=1+\frac{N}{N-2}(z+z^{N-1})+z^N$ and some univalent polynomials

Key Result

Theorem 1

All zeros of the polynomial lie on the unit circle if and only if the polynomial $P(z)$ is self-reciprocal---which means, for a polynomial with real coeficients, that $P(z)=\pm z^N P(1/z)$---and all zeros of the polynomial $P'(z)$ belong to the closed central unit disk $\overline{\mathbb{D}}=\{z:|z|\le1\}$.

Figures (4)

  • Figure 1: Stability domains of the trinomial $f(z)=z^n+az^{n-1}+b$ in the $(a,b)$-plane: i) corresponds to the case of even $n$, ii) is for odd $n$.
  • Figure 2: Stability domains of the trinomial $f(z)=z^n+az^{n-1}+b$ in the $(a,b)$-plane: i) even $n$, ii) odd $n$. The line $b=a/n$ is in bold, the line $b=-a/n$ is dashed.
  • Figure 3: Location of the points $(-1,0)$ and $(\gamma_j,\pm\sqrt{1-(\gamma_j)^2})$, $j=1,\ldots,(N-3)/2=4$ for $N=11$, on the unit circle.
  • Figure 4: Graphs of the images of the unit circle: i) under the mappings $F_{11}^{(1)}(z)$ and $F_{12}^{(3)}(z)$ (dashed); ii) under the mappings $F_{11}^{(2)}(z)$ and $F_{12}^{(4)}(z)$ (dashed).

Theorems & Definitions (16)

  • Theorem 1: AC22
  • Theorem 2: SK94
  • Theorem 3: TS72
  • Theorem 4
  • proof
  • Theorem 5
  • proof : Proof
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 6 more