{
  "id": "1803.02775",
  "version": "v1",
  "published": "2018-03-07T17:14:21.000Z",
  "updated": "2018-03-07T17:14:21.000Z",
  "title": "Dualities in the $q$-Askey scheme and degenerated DAHA",
  "authors": [
    "Tom H. Koornwinder",
    "Marta Mazzocco"
  ],
  "comment": "44 pages, 1 figure",
  "categories": [
    "math.CA",
    "math.QA",
    "nlin.SI"
  ],
  "abstract": "The Askey-Wilson polynomials are a four-parameter family of orthogonal symmetric Laurent polynomials $R_n[z]$ which are eigenfunctions of a second-order $q$-difference operator $L$, and of a second-order difference operator in the variable $n$ with eigenvalue $z + z^{-1}=2x$. Then $L$ and multiplication by $z+z^{-1}$ generate the Askey-Wilson (Zhedanov) algebra. A nice property of the Askey-Wilson polynomials is that the variables $z$ and $n$ occur in the explicit expression in a similar and to some extent exchangeable way. This property is called duality. It returns in the non-symmetric case and in the underlying algebraic structures: the Askey-Wilson algebra and the double affine Hecke algebra (DAHA). In this paper we follow the degeneration of the Askey-Wilson polynomials until two arrows down and in four diferent situations: for the orthogonal polynomials themselves, for the degenerate Askey-Wilson algebras, for the non-symmetric polynomials and for the (degenerate) DAHA and its representations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-07T17:14:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33D45",
      "33D80",
      "33D52",
      "16T99"
    ],
    "keywords": [
      "askey scheme",
      "degenerated daha",
      "askey-wilson polynomials",
      "orthogonal symmetric laurent polynomials",
      "second-order difference operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}