{
  "id": "1904.08283",
  "version": "v1",
  "published": "2019-04-16T17:23:32.000Z",
  "updated": "2019-04-16T17:23:32.000Z",
  "title": "Inversion formula with hypergeometric polynomials and its application to an integral equation",
  "authors": [
    "Ridha Nasri",
    "Alain Simonian",
    "Fabrice Guillemin"
  ],
  "comment": "22 pages, no figure",
  "categories": [
    "math.CA",
    "cs.DM",
    "cs.PF"
  ],
  "abstract": "For any complex parameters $x$ and $\\nu$, we provide a new class of linear inversion formulas $T = A(x,\\nu) \\cdot S \\Leftrightarrow S = B(x,\\nu) \\cdot T$ between sequences $S = (S_n)_{n \\in \\mathbb{N}^*}$ and $T = (T_n)_{n \\in \\mathbb{N}^*}$, where the infinite lower-triangular matrix $A(x,\\nu)$ and its inverse $B(x,\\nu)$ involve Hypergeometric polynomials $F(\\cdot)$, namely $$ \\left\\{ \\begin{array}{ll} A_{n,k}(x,\\nu) = \\displaystyle (-1)^k\\binom{n}{k}F(k-n,-n\\nu;-n;x), \\\\ B_{n,k}(x,\\nu) = \\displaystyle (-1)^k\\binom{n}{k}F(k-n,k\\nu;k;x) \\end{array} \\right. $$ for $1 \\leqslant k \\leqslant n$. Functional relations between the ordinary (resp. exponential) generating functions of the related sequences $S$ and $T$ are also given. These new inversion formulas have been initially motivated by the resolution of an integral equation recently appeared in the field of Queuing Theory; we apply them to the full resolution of this integral equation. Finally, matrices involving generalized Laguerre polynomials polynomials are discussed as specific cases of our general inversion scheme.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-16T17:23:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "45B99",
      "33C05",
      "15B99"
    ],
    "keywords": [
      "integral equation",
      "hypergeometric polynomials",
      "application",
      "linear inversion formulas",
      "infinite lower-triangular matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}