{
  "id": "math/9307218",
  "version": "v1",
  "published": "1993-07-09T00:00:00.000Z",
  "updated": "1993-07-09T00:00:00.000Z",
  "title": "Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials.",
  "authors": [
    "Alphonse P. Magnus"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w'/w$ is a rational function) are shown to be solutions of non linear differential equations with respect to a well-chosen parameter, according to principles established by D. G. Chudnovsky. Examples are given. For instance, the recurrence coefficients in $a_{n+1}p_{n+1}(x)=xp_n(x) -a_np_{n-1}(x)$ of the orthogonal polynomials related to the weight $\\exp(-x^4/4-tx^2)$ on {\\blackb R\\/} satisfy $4a_n^3\\ddot a_n = (3a_n^4+2ta_n^2-n)(a_n^4+2ta_n^2+n)$, and $a_n^2$ satisfies a Painlev\\'e ${\\rm P}_{\\rm IV}$ equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1993-07-09T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semi-classical orthogonal polynomials",
      "recurrence coefficients",
      "painlevé-type differential equations",
      "non linear differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1993math......7218M"
    }
  }
}