{
  "id": "1309.4237",
  "version": "v2",
  "published": "2013-09-17T09:13:04.000Z",
  "updated": "2013-10-17T08:13:00.000Z",
  "title": "Positivity properties of Jacobi-Stirling numbers and generalized Ramanujan polynomials",
  "authors": [
    "Zhicong Lin",
    "Jiang Zeng"
  ],
  "comment": "17 pages, 2 tables, the proof of Lemma 3.3 is corrected, final version to appear in Advances in Applied Mathematics",
  "categories": [
    "math.CO"
  ],
  "abstract": "Generalizing recent results of Egge and Mongelli, we show that each diagonal sequence of the Jacobi-Stirling numbers $\\js(n,k;z)$ and $\\JS(n,k;z)$ is a P\\'olya frequency sequence if and only if $z\\in [-1, 1]$ and study the $z$-total positivity properties of these numbers. Moreover, the polynomial sequences $$\\biggl\\{\\sum_{k=0}^n\\JS(n,k;z)y^k\\biggr\\}_{n\\geq 0}\\quad \\text{and} \\quad \\biggl\\{\\sum_{k=0}^n\\js(n,k;z)y^k\\biggr\\}_{n\\geq 0}$$ are proved to be strongly $\\{z,y\\}$-log-convex. In the same vein, we extend a recent result of Chen et al. about the Ramanujan polynomials to Chapoton's generalized Ramanujan polynomials. Finally, bridging the Ramanujan polynomials and a sequence arising from the Lambert $W$ function, we obtain a neat proof of the unimodality of the latter sequence, which was proved previously by Kalugin and Jeffrey.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-17T08:13:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05A20",
      "12D10"
    ],
    "keywords": [
      "jacobi-stirling numbers",
      "polya frequency sequence",
      "total positivity properties",
      "chapotons generalized ramanujan polynomials",
      "neat proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.4237L"
    }
  }
}