{
  "id": "1701.07016",
  "version": "v1",
  "published": "2017-01-24T03:36:08.000Z",
  "updated": "2017-01-24T03:36:08.000Z",
  "title": "Factors of sums involving $q$-binomial coefficients and powers of $q$-integers",
  "authors": [
    "Victor J. W. Guo",
    "Su-Dan Wang"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We show that, for all positive integers $n_1, \\ldots, n_m$, $n_{m+1}=n_1$, and any non-negative integers $j$ and $r$ with $j\\leqslant m$, the expression $$ \\frac{1}{[n_1]}{n_1+n_{m}\\brack n_1}^{-1} \\sum_{k=1}^{n_1}[2k][k]^{2r}q^{jk^2-(r+1)k}\\prod_{i=1}^{m} {n_i+n_{i+1}\\brack n_i+k} $$ is a Laurent polynomial in $q$ with integer cofficients, where $[n]=1+q+\\cdots+q^{n-1}$ and ${n\\brack k}=\\prod_{i=1}^k(1-q^{n-i+1})/(1-q^i)$. This gives a $q$-analogue of a divisibility result on the Catalan triangle obtained by the first author and Zeng, and also confirms a conjecture of the first author and Zeng. We further propose several related conjectures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-24T03:36:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A30",
      "05A10",
      "11B65"
    ],
    "keywords": [
      "binomial coefficients",
      "first author",
      "integer cofficients",
      "divisibility result",
      "catalan triangle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}