{
  "id": "2103.08303",
  "version": "v1",
  "published": "2021-03-15T12:03:28.000Z",
  "updated": "2021-03-15T12:03:28.000Z",
  "title": "Weighted $L^2$-Norms of Gegenbauer polynomials",
  "authors": [
    "Johann S. Brauchart",
    "Peter J. Grabner"
  ],
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "We study integrals of the form \\begin{equation*} \\int_{-1}^1(C_n^{(\\lambda)}(x))^2(1-x)^\\alpha (1+x)^\\beta\\, dx, \\end{equation*} where $C_n^{(\\lambda)}$ denotes the Gegenbauer-polynomial of index $\\lambda>0$ and $\\alpha,\\beta>-1$. We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as $n\\to\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-15T12:03:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "33C20",
      "41A60"
    ],
    "keywords": [
      "gegenbauer polynomials",
      "study integrals",
      "exact formulas",
      "asymptotic formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}