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 \[P(z) = \sum_{j = 0}^{s}(-1)^jγ_j\left(z^j - z^{N + s + 1 - j}\right), \ γ_0 = 1.\] It is shown that if all the zeros of $P(z)$ are located on the unit circle, that $\left|{\hat{I}}_j^3\right|\le \left(\genfrac{}{}{0ex}{}{s}{j}\right){\left(\left(\genfrac{}{}{0ex}{}{N+s+1}{j}\right)\right)}^{-1}$, $j=1,\dots ,s$; moreover, these estimates cannot be improved in the general case. Factorization formulas for extremal polynomials are given: \[ \begin{align} \phantom{a} & \sum_{j = 0}^{s}(-1)^j{s \choose j}\left({N + s + 1 \choose j}\right)^{-1}\left(z^j - z^{N + s + 1 - j}\right) \\ &= (1 - z)^{2s + 1} \prod_{j = 1}^{\left[\frac{N - s}{2}\right]} \left[z^2 + 1 + 2z(1 - 2(ν_j)^2)\right] \begin{cases} (1 + z), & N - s \mbox{ is odd} \\ 1, & N - s \mbox{ is even} \end{cases} \end{align} \] where ${\left\{\hat{I}{½}_j\right\}}_{j=1}^{\left[\frac{N-s}{2}\right]}$ is the set of positive zeros of the polynomial $U_N^{(s)}(z)$ given $U_N(z)=\sum _{j=0}^{\left[\frac{N}{2}\right]}(-1{)}^j\frac{(N-j)!}{j!(N-2j)!}(2z{)}^{N-2j}$ 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: \[\frac{2^s}{s!}(1 - z^2)^sU_N^{(s)}(z) = (-1)^s \sum_{j = 0}^{s}(-1)^j{N-j \choose N-s} {N+s+1 \choose j} U_{N + s - 2j}(z). \]