{
  "id": "2308.13583",
  "version": "v1",
  "published": "2023-08-25T15:17:03.000Z",
  "updated": "2023-08-25T15:17:03.000Z",
  "title": "Orthogonality of the big $-1$ Jacobi polynomials for non-standard parameters",
  "authors": [
    "Howard S. Cohl",
    "Roberto S. Costas-Santos"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "The big $-1$ Jacobi polynomials $(Q_n^{(0)}(x;\\alpha,\\beta,c))_n$ have been classically defined for $\\alpha,\\beta\\in(-1,\\infty)$, $c\\in(-1,1)$. We extend this family so that wider sets of parameters are allowed, i.e., they are non-standard. Assuming initial conditions $Q^{(0)}_0(x)=1$, $Q^{(0)}_{-1}(x)=0$, we consider the big $-1$ Jacobi polynomials as monic orthogonal polynomials which therefore satisfy the following three-term recurrence relation \\[ xQ^{(0)}_n(x)=Q^{(0)}_{n+1}(x)+b_{n} Q^{(0)}_n(x)+ u_{n} Q^{(0)}_{n-1}(x), \\quad n=0, 1, 2,\\ldots. \\] For standard parameters, the coefficients $u_n>0$ for all $n$. We discuss the situation where Favard's theorem cannot be directly applied for some positive integer $n$ such that $u_n=0$. We express the big $-1$ Jacobi polynomials for non-standard parameters as a product of two polynomials. Using this factorization, we obtain a bilinear form with respect to which these polynomials are orthogonal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-25T15:17:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "44A20",
      "42C05",
      "33C05"
    ],
    "keywords": [
      "jacobi polynomials",
      "non-standard parameters",
      "orthogonality",
      "monic orthogonal polynomials",
      "three-term recurrence relation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}