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

TL;DR

This paper investigates reciprocal antisymmetric polynomials with zeros on the unit circle, establishes bounds on their coefficients, provides factorization formulas involving derivatives of Chebyshev polynomials of the second kind, and expresses these derivatives as linear combinations of Chebyshev polynomials.

Contribution

It introduces new bounds for coefficients of reciprocal polynomials with zeros on the unit circle and derives explicit formulas connecting derivatives of Chebyshev polynomials to linear combinations of Chebyshev polynomials.

Findings

Bounds on polynomial coefficients are proven to be optimal.

Factorization formulas involve derivatives of Chebyshev polynomials.

Explicit linear combination formulas for derivatives of Chebyshev polynomials are obtained.

Abstract

In this article, we consider the reciprocal antisymmetric polynomial \[P(z) = \sum_{j = 0}^{s}(-1)^j\gamma_j\left(z^j - z^{N + s + 1 - j}\right), \ \gamma_0 = 1.\] It is shown that if all the zeros of $P(z)$ are located on the unit circle, that $\displaystyle\left|\gamma_j\right| \leq {s \choose j}\left({N + s + 1 \choose j}\right)^{-1}$, $j = 1,\ldots,s$; moreover, these estimates cannot be improved in the general case. Factorization formulas for extremal polynomials are given: \[ \begin{align} \phantom{a} & \sum_{j = 0}^{s}(-1)^j{s \choose j}\left({N + s + 1 \choose j}\right)^{-1}\left(z^j - z^{N + s + 1 - j}\right) \\ &= (1 - z)^{2s + 1} \prod_{j = 1}^{\left[\frac{N - s}{2}\right]} \left[z^2 + 1 + 2z(1 - 2(\nu_j)^2)\right] \begin{cases} (1 + z), & N - s \mbox{ is odd} \\ 1, & N - s \mbox{ is even} \end{cases} \end{align} \] where $\left\{\nu_j\right\}_{j = 1}^{\left[\frac{N - s}{2}\right]}$ is the set of positive zeros of the polynomial $U_N^{(s)}(z)$ given $\displaystyle U_N(z) = \sum_{j = 0}^{\left[\frac{N}{2}\right]} (-1)^j \frac{(N - j)!}{j!(N - 2j)!}(2z)^{N - 2j}$ 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: \[\frac{2^s}{s!}(1 - z^2)^sU_N^{(s)}(z) = (-1)^s \sum_{j = 0}^{s}(-1)^j{N-j \choose N-s} {N+s+1 \choose j} U_{N + s - 2j}(z). \]