{
  "id": "0910.5667",
  "version": "v6",
  "published": "2009-10-29T15:56:06.000Z",
  "updated": "2010-06-15T06:49:28.000Z",
  "title": "On sums of binomial coefficients modulo p^2",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "13 pages, polished version",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum $\\sum_{k=0}^{p^a-1}\\binom{hp^a-1}{k}\\binom{2k}{k}/m^k$ mod p^2, where h,m are p-adic integers with m\\not=0 (mod p). For example, we show that if h\\not=0 (mod p) and p^a>3 then $$ sum_{k=0}^{p^a-1}\\binom{hp^a-1}{k}\\binom{2k}{k}(-h/2)^k =(\\frac{1-2h}{p^a})(1+h((4h-2)^{p-1}/h^{p-1}-1)) (mod p^2),$$ where (-) denotes the Jacobi symbol. Here is another remarkable congruence: If p>3 then $$\\sum_{k=0}^{p^a-1}\\binom{p^a-1}{k}\\binom{2k}{k}(-1)^k =3^{p-1}(\\frac{p^a}3) (mod p^2).$$",
  "revisions": [
    {
      "version": "v6",
      "updated": "2010-06-15T06:49:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "05A10",
      "11A07",
      "11S99"
    ],
    "keywords": [
      "binomial coefficients modulo",
      "jacobi symbol",
      "odd prime",
      "p-adic integers",
      "positive integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0910.5667S"
    }
  }
}