{
  "id": "1405.2812",
  "version": "v2",
  "published": "2014-05-12T15:39:21.000Z",
  "updated": "2014-06-29T17:12:42.000Z",
  "title": "An integral identity with applications in orthogonal polynomials",
  "authors": [
    "Yuan Xu"
  ],
  "comment": "Correct mix-up of parameters in the proof of Theorem 3.4 and add an appendix on the original proof of Theorem 1.1",
  "categories": [
    "math.CA"
  ],
  "abstract": "For $\\boldsymbol{\\large {\\lambda}} = (\\lambda_1,\\ldots,\\lambda_d)$ with $\\lambda_i > 0$, it is proved that \\begin{equation*} \\prod_{i=1}^d \\frac{ 1}{(1- r x_i)^{\\lambda_i}} = \\frac{\\Gamma(|\\boldsymbol{\\large {\\lambda}}|)}{\\prod_{i=1}^{d} \\Gamma(\\lambda_i)} \\int_{\\mathcal{T}^d} \\frac{1}{ (1- r \\langle x, u \\rangle)^{|\\boldsymbol{\\large {\\lambda}}|}} \\prod_{i=1}^d u_i^{\\lambda_i-1} du, \\end{equation*} where $\\mathcal{T}^d$ is the simplex in homogeneous coordinates of $\\mathbb{R}^d$, from which a new integral relation for Gegenbuer polynomials of different indexes is deduced. The latter result is used to derive closed formulas for reproducing kernels of orthogonal polynomials on the unit cube and on the unit ball.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-29T17:12:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "33C50",
      "42C10"
    ],
    "keywords": [
      "orthogonal polynomials",
      "integral identity",
      "applications",
      "unit ball",
      "unit cube"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.2812X"
    }
  }
}