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

Abstract

We show that all the zeros of the quadrinomial $p(z)=1+\kappa (z+z^{N-1})+z^N$ 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 $q(z)=1+\kappa (z-z^{N-1})-z^N$, 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 $\kappa$, we provide factorization formulas for the corresponding quadrinomials. For example, when $N$ is odd and $\kappa =N/(N-2)$, the following representation is valid: \[ p(z)=(1+z)^3\prod_{j=1}^{(N-3)/2}[1+z^2-2zγ_j], \] where ${\gamma }_j=1-2{\nu }_j^2$ with $\{{\nu }_j{\}}_{j=1}^{(N-3)/2}$ being the collection of positive roots of the equation $U_{N-2}^{\prime }(x)=0$; 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 $U_j^{\prime }(x)$ 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 $q(z)$.