{
  "id": "2410.14009",
  "version": "v1",
  "published": "2024-10-17T20:22:00.000Z",
  "updated": "2024-10-17T20:22:00.000Z",
  "title": "Some properties of the quadrinomials $p(z)=1+κ(z+z^{N-1})+z^N$ and $q(z)=1+κ(z-z^{N-1})-z^N$",
  "authors": [
    "Dmitriy Dmitrishin",
    "Alexander Stokolos"
  ],
  "categories": [
    "math.CA"
  ],
  "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\\le\\kappa\\le 1\\; (\\mbox{ if $N$ is even}),\\;\\; -1\\le\\kappa\\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)\\le\\kappa\\le 1\\; (\\text{ if $N$ is odd}),\\;\\; -N/(N-2)\\le\\kappa\\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^2-2z\\gamma_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 $U'_{N-2}(x)=0$; 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 $U'_j(x)$ 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 Fej\\'er polynomial, as well as construct certain univalent polynomials related to the polynomials $p(z)$ and $q(z)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-17T20:22:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26C10"
    ],
    "keywords": [
      "properties",
      "similar factorization formulas",
      "inequalities",
      "second kind",
      "chebyshev polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}