{
  "id": "0909.5648",
  "version": "v12",
  "published": "2009-09-30T19:57:32.000Z",
  "updated": "2016-02-15T16:20:00.000Z",
  "title": "Binomial coefficients, Catalan numbers and Lucas quotients",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "24 pages. Correct few typos",
  "journal": "Sci. China Math. 53(2010), no.9, 2473-2488",
  "doi": "10.1007/s11425-010-3151-3",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $p$ be an odd prime and let $a,m$ be integers with $a>0$ and $m \\not\\equiv0\\pmod p$. In this paper we determine $\\sum_{k=0}^{p^a-1}\\binom{2k}{k+d}/m^k$ mod $p^2$ for $d=0,1$; for example, $$\\sum_{k=0}^{p^a-1}\\frac{\\binom{2k}k}{m^k}\\equiv\\left(\\frac{m^2-4m}{p^a}\\right)+\\left(\\frac{m^2-4m}{p^{a-1}}\\right)u_{p-(\\frac{m^2-4m}{p})}\\pmod{p^2},$$ where $(-)$ is the Jacobi symbol, and $\\{u_n\\}_{n\\geqslant0}$ is the Lucas sequence given by $u_0=0$, $u_1=1$ and $u_{n+1}=(m-2)u_n-u_{n-1}$ for $n=1,2,3,\\ldots$. As an application, we determine $\\sum_{0<k<p^a,\\, k\\equiv r\\pmod{p-1}}C_k$ modulo $p^2$ for any integer $r$, where $C_k$ denotes the Catalan number $\\binom{2k}k/(k+1)$. We also pose some related conjectures.",
  "revisions": [
    {
      "version": "v11",
      "updated": "2010-08-31T15:39:41.000Z",
      "abstract": "Let p be an odd prime and let a,m be integers with a>0 and m not=0 (mod p). In this paper we determine $\\sum_{k=0}^{p^a-1}\\binom[2k,k+d]/m^k$ mod p^2 for d=0,1; for example, $$\\sum_{k=0}^{p^a-1}\\binom[2k,k]/m^k=(\\frac{m^2-4m}{p^a})+(\\frac{m^2-4m}{p^{a-1}})u_{p-(\\frac{m^2-4m}{p})} (mod p^2),$$ where (-) is the Jacobi symbol, u_0=0, u_1=1 and u_{n+1}=(m-2)u_n-u_{n-1} for n=1,2,3,.... As an application, we determine $\\sum_{0<k<p^a, k=r(mod p-1)}C_k$ modulo p^2 for any integer r, where $C_k$ denotes the Catalan number $\\binom[2k,k]/(k+1)$. We also pose some related conjectures.",
      "comment": "24 pages",
      "doi": null
    },
    {
      "version": "v12",
      "updated": "2016-02-15T16:20:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "05A10",
      "11A07"
    ],
    "keywords": [
      "catalan number",
      "lucas quotients",
      "binomial coefficients",
      "jacobi symbol",
      "odd prime"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s11425-010-3151-3",
      "journal": "Science in China A: Mathematics",
      "year": 2010,
      "month": "Jun",
      "volume": 53,
      "number": 9,
      "pages": 2473,
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010ScChA..53.2473S"
    }
  }
}