{
  "id": "2204.01008",
  "version": "v1",
  "published": "2022-04-03T06:50:49.000Z",
  "updated": "2022-04-03T06:50:49.000Z",
  "title": "Turán inequalities from Chebyshev to Laguerre polynomials",
  "authors": [
    "Bernhard Heim",
    "Markus Neuhauser",
    "Robert Troeger"
  ],
  "categories": [
    "math.CA",
    "math.NT"
  ],
  "abstract": "Let $g$ and $h$ be real-valued arithmetic functions, positive and normalized. Specific choices within the following general scheme of recursively defined polynomials \\begin{equation*} P_n^{g,h}(x):= \\frac{x}{h(n)} \\sum_{k=1}^{n} g(k) \\, P_{n-k}^{g,h}(x), \\end{equation*} with initial value $P_{0}^{g,h}(x)=1$ encode information about several classical, widely studied polynomials. This includes Chebyshev polynomials of the second kind, associated Laguerre polynomials, and the Nekrasov--Okounkov polynomials. In this paper we prove that for $g(n)=n$ and fixed $h$ we obtain orthogonal polynomial sequences for positive definite functionals. Let $h(n)=n^s$ with $0 \\leq s \\leq 1 $. Then the sequence satisfies Tur\\'an inequalities for $x \\geq 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-03T06:50:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "laguerre polynomials",
      "turán inequalities",
      "sequence satisfies turan inequalities",
      "orthogonal polynomial sequences",
      "real-valued arithmetic functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}