{
  "id": "math/9511218",
  "version": "v1",
  "published": "1995-11-02T00:00:00.000Z",
  "updated": "1995-11-02T00:00:00.000Z",
  "title": "Contiguous relations, continued fractions and orthogonality",
  "authors": [
    "Dharma P. Gupta",
    "David R. Masson"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We examine a special linear combination of balanced very-well-poised $\\tphia$ basic hypergeometric series that is known to satisfy a transformation. We call this $\\Phi$ and show that it satisfies certain three-term contiguous relations. From two sets of contiguous relations for $\\Phi$ we obtain fifty-six pairwise linearly independent solutions to a three-term recurrence that generalizes the recurrence for Askey-Wilson polynomials. The associated continued fraction is evaluated using Pincherle's theorem. From this continued fraction we are able to derive a discrete system of biorthogonal rational functions. This ties together Wilson's results for rational biorthogonality, Watson's $q$-analogue of Ramanujan's Entry 40 continued fraction and a conjecture of Askey concerning the latter. Some new $q$-series identities are also obtained. One is an important three-term transformation for $\\Phi$'s which generalizes all the known two and three-term $\\ephis$ transformations. Others are new and unexpected quadratic identities for these very-well-poised $\\ephis$'s.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1995-11-02T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continued fraction",
      "orthogonality",
      "biorthogonal rational functions",
      "special linear combination",
      "important three-term transformation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1995math.....11218G"
    }
  }
}