{
  "id": "1908.07737",
  "version": "v1",
  "published": "2019-08-21T07:48:48.000Z",
  "updated": "2019-08-21T07:48:48.000Z",
  "title": "Some results on vanishing coefficients in infinite product expansions",
  "authors": [
    "Nayandeep Deka Baruah",
    "Mandeep Kaur"
  ],
  "comment": "15 pages",
  "journal": "The Ramanujan Journal, 2019",
  "doi": "10.1007/s11139-019-00172-x",
  "categories": [
    "math.NT"
  ],
  "abstract": "Recently, M. D. Hirschhorn proved that, if $\\sum_{n=0}^\\infty a_nq^n := (-q,-q^4;q^5)_\\infty(q,q^9;q^{10})_\\infty^3$ and $\\sum_{n=0}^\\infty b_nq^n:=(-q^2,-q^3;q^5)_\\infty(q^3,q^7;q^{10})_\\infty^3$, then $a_{5n+2}=a_{5n+4}=0$ and $b_{5n+1}=b_{5n+4}=0$. Motivated by the work of Hirschhorn, D. Tang proved some comparable results including the following: If $ \\sum_{n=0}^\\infty c_nq^n := (-q,-q^4;q^5)_\\infty^3(q^3,q^7;q^{10})_\\infty$ and $\\sum_{n=0}^\\infty d_nq^n := (-q^2,-q^3;q^5)_\\infty^3(q,q^9;q^{10})_\\infty$, then $c_{5n+3}=c_{5n+4}=0$ and $d_{5n+3}=d_{5n+4}=0$. In this paper, we prove that $a_{5n}=b_{5n+2}$, $a_{5n+1}=b_{5n+3}$, $a_{5n+2}=b_{5n+4}$, $a_{5n-1}=b_{5n+1}$, $c_{5n+3}=d_{5n+3}$, $c_{5n+4}=d_{5n+4}$, $c_{5n}=d_{5n}$, $c_{5n+2}=d_{5n+2}$, and $c_{5n+1}>d_{5n+1}$. We also record some other comparable results not listed by Tang.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-21T07:48:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33D15",
      "11F33"
    ],
    "keywords": [
      "infinite product expansions",
      "vanishing coefficients",
      "comparable results",
      "hirschhorn"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}