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Reciprocal Polynomials with Zeros on the Unit Circle and Derivatives of Chebyshev Polynomials of the Second Kind

Dmitriy Dmitrishin, Daniel Gray, Alexander Stokolos

Abstract

In this article, we consider the reciprocal antisymmetric polynomial \begin{displaymath} P(z) = \sum_{j = 0}^{s}(-1)^jγ_j\pa{z^j - z^{N + s + 1 - j}}, \ γ_0 = 1. \end{displaymath} It is shown that if all the zeros of $P(z)$ are located on the unit circle, that $\displaystyle\abs{γ_j} \leq {s \choose j}\pa{N + s + 1 \choose j}^{-1}$, $j = 1,\ldots,s$; moreover, these estimates cannot be improved in the general case. Factorization formulas for extremal polynomials are given: \begin{eqnarray*} \lefteqn{\sum_{j = 0}^{s}(-1)^j{s \choose j}\pa{N + s + 1 \choose j}^{-1}\pa{z^j - z^{N + s + 1 - j}}} \\ &=& (1 - z)^{2s + 1} \prod_{j = 1}^{\br{\frac{N - s}{2}}} \br{z^2 + 1 + 2z\pa{1 - 2(ν_j)^2}} \begin{cases} (1 + z), & N - s \mbox{ is odd} \\ 1, & N - s \mbox{ is even} \end{cases} \end{eqnarray*} where $\cb{ν_j}_{j = 1}^{\br{\frac{N - s}{2}}}$ is the set of positive zeros of the polynomial $U_N^{(s)}(z)$ given $\displaystyle U_N(z) = \sum_{j = 0}^{\br{\frac{N}{2}}} (-1)^j \frac{(N - j)!}{j!(N - 2j)!}(2z)^{N - 2j}$ are the Chebyshev Polynomials of the Second Kind and $U_N^{(s)}(z)$ is the $s$th derivative of $U_N(z)$. As an application of the results, formulas were obtained expressing the derivatives of Chebyshev polynomials of the second kind through linear combinations of Chebyshev polynomials of the second kind: \begin{displaymath} \frac{2^s}{s!}(1 - z^2)^sU_N^{(s)}(z) = (-1)^s \sum_{j = 0}^{s}(-1)^j{N-j \choose N-s} {N+s+1 \choose j} U_{N + s - 2j}(z). \end{displaymath}

Reciprocal Polynomials with Zeros on the Unit Circle and Derivatives of Chebyshev Polynomials of the Second Kind

Abstract

In this article, we consider the reciprocal antisymmetric polynomial \begin{displaymath} P(z) = \sum_{j = 0}^{s}(-1)^jγ_j\pa{z^j - z^{N + s + 1 - j}}, \ γ_0 = 1. \end{displaymath} It is shown that if all the zeros of are located on the unit circle, that , ; moreover, these estimates cannot be improved in the general case. Factorization formulas for extremal polynomials are given: \begin{eqnarray*} \lefteqn{\sum_{j = 0}^{s}(-1)^j{s \choose j}\pa{N + s + 1 \choose j}^{-1}\pa{z^j - z^{N + s + 1 - j}}} \\ &=& (1 - z)^{2s + 1} \prod_{j = 1}^{\br{\frac{N - s}{2}}} \br{z^2 + 1 + 2z\pa{1 - 2(ν_j)^2}} \begin{cases} (1 + z), & N - s \mbox{ is odd} \\ 1, & N - s \mbox{ is even} \end{cases} \end{eqnarray*} where is the set of positive zeros of the polynomial given are the Chebyshev Polynomials of the Second Kind and is the th derivative of . As an application of the results, formulas were obtained expressing the derivatives of Chebyshev polynomials of the second kind through linear combinations of Chebyshev polynomials of the second kind: \begin{displaymath} \frac{2^s}{s!}(1 - z^2)^sU_N^{(s)}(z) = (-1)^s \sum_{j = 0}^{s}(-1)^j{N-j \choose N-s} {N+s+1 \choose j} U_{N + s - 2j}(z). \end{displaymath}
Paper Structure (6 sections, 10 theorems, 48 equations, 1 figure)

This paper contains 6 sections, 10 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Main Results
  4. Extremal properties of polynomials with zeros on the unit circle
  5. Some examples
  6. Representation of derivatives of the Chebyshev polynomial of the second kind in terms of Chebyshev polynomials of the second kind

Key Result

Theorem 1.1

ref2. All zeros of the polynomial $P(z) = \sum_{j = 0}^{N}a_j z^{N - j}$ with $a_0 \neq 0$ lie on the unit circle if and only if the polynomial $P(z)$ is reciprocal and all zeros of the polynomial $P'(z)$ belong to the closed central unit disc $\overline{\mathbb D} = \{z: \left|z\right| \leq 1\}$.

Figures (1)

  • Figure 1: Sets in the plane of coefficients $\gamma_1, \ \gamma_2$ for polynomial \ref{['eqn8']}, in which all zeros of this polynomial belong to the unit circle for $N = 11$ (a) and $N = 12$ (b). The dashed lines shows the boundaries of the parallelepiped $\hat{\Pi}$ and cube $\tilde{\Pi}$ while the solid lines show the boundaries formed by $U_0$, $U_{\pi}$, and $\bigcup_{\tau \in (0,\pi)} \{U_{\tau}\}$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • ...and 7 more