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

In this article, we consider the reciprocal antisymmetric polynomial $$P(z)=\sum _{j=0}^s(-1{)}^j{\gamma }_j(z^j-z^{N+s+1-j}),{\gamma }_0=1.$$ It is shown that if all the zeros of $P(z)$ are located on the unit circle, that ${\displaystyle \mid {\gamma }_j\mid \le \left(\genfrac{}{}{0px}{}{s}{j}\right){\left(\left(\genfrac{}{}{0px}{}{N+s+1}{j}\right)\right)}^{-1}}$, $j=1,\dots ,s$; moreover, these estimates cannot be improved in the general case. Factorization formulas for extremal polynomials are given: $$\begin{array}{rllll}& {\displaystyle \phantom{a}}& {\displaystyle \sum _{j=0}^s(-1{)}^j\left(\genfrac{}{}{0px}{}{s}{j}\right){\left(\left(\genfrac{}{}{0px}{}{N+s+1}{j}\right)\right)}^{-1}(z^j-z^{N+s+1-j})}& & \\ & {\displaystyle }& {\displaystyle =(1-z{)}^{2s+1}\prod _{j=1}^{\left[\frac{N-s}{2}\right]}[z^2+1+2z(1-2({\nu }_j{)}^2)]\{\begin{array}{ll}(1+z),& N-sisodd\\ 1,& N-siseven\end{array}}& & \end{array}$$ 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: $$\frac{2^s}{s!}(1-z^2{)}^s{U}_N^{(s)}(z)=(-1{)}^s\sum _{j=0}^s(-1{)}^j\left(\genfrac{}{}{0px}{}{N-j}{N-s}\right)\left(\genfrac{}{}{0px}{}{N+s+1}{j}\right)U_{N+s-2j}(z)\mathrm{.}$$