{
  "id": "2101.04479",
  "version": "v1",
  "published": "2021-01-12T13:53:21.000Z",
  "updated": "2021-01-12T13:53:21.000Z",
  "title": "On the generalized hypergeometric function, Sobolev orthogonal polynomials and biorthogonal rational functions",
  "authors": [
    "Sergey M. Zagorodnyuk"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "It turned out that the partial sums $g_n(z) = \\sum_{k=0}^n \\frac{(a_1)_k ... (a_p)_k}{(b_1)_k ... (b_q)_k} \\frac{z^k}{k!}$, of the generalized hypergeometric series ${}_p F_q(a_1,...,a_p; b_1,...,b_q;z)$, with parameters $a_j,b_l\\in\\mathbb{C}\\backslash\\{ 0,-1,-2,... \\}$, are Sobolev orthogonal polynomials. The corresponding monic polynomials $G_n(z)$ are polynomials of $R_I$ type, and therefore they are related to biorthogonal rational functions. Polynomials $g_n$ possess a differential equation (in $z$), and a recurrence relation (in $n$). We study integral representations for $g_n$, and some other their basic properties. Partial sums of arbitrary power series with non-zero coefficients are shown to be also related to biorthogonal rational functions. We obtain a relation of polynomials $g_n(z)$ to Jacobi-type pencils and their associated polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-12T13:53:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C05"
    ],
    "keywords": [
      "biorthogonal rational functions",
      "sobolev orthogonal polynomials",
      "generalized hypergeometric function",
      "partial sums",
      "study integral representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}