{
  "id": "2208.06483",
  "version": "v1",
  "published": "2022-08-12T20:04:29.000Z",
  "updated": "2022-08-12T20:04:29.000Z",
  "title": "On orthogonal Laurent polynomials related to the partial sums of power series",
  "authors": [
    "Sergey M. Zagorodnyuk"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $f(z) = \\sum_{k=0}^\\infty d_k z^k$, $d_k\\in\\mathbb{C}\\backslash\\{ 0 \\}$, $d_0=1$, be a power series with a non-zero radius of convergence $\\rho$: $0 <\\rho \\leq +\\infty$. Denote by $f_n(z)$ the n-th partial sum of $f$, and $R_{2n}(z) = \\frac{ f_{2n}(z) }{ z^n }$, $R_{2n+1}(z) = \\frac{ f_{2n+1}(z) }{ z^{n+1} }$, $n=0,1,2,...$. By the result of Hendriksen and Van Rossum there exists a linear functional $\\mathbf{L}$ on Laurent polynomials, such that $\\mathbf{L}(R_n R_m) = 0$, when $n\\not= m$, while $\\mathbf{L}(R_n^2)\\not= 0$. We present an explicit integral representation for $\\mathbf{L}$ in the above case of the partial sums. We use methods from the theory of generating functions. The case of finite systems of such Laurent polynomials is studied as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-12T20:04:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C05"
    ],
    "keywords": [
      "orthogonal laurent polynomials",
      "power series",
      "n-th partial sum",
      "explicit integral representation",
      "van rossum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}