{
  "id": "1803.01830",
  "version": "v1",
  "published": "2018-03-05T18:50:28.000Z",
  "updated": "2018-03-05T18:50:28.000Z",
  "title": "A $q$-microscope for supercongruences",
  "authors": [
    "Victor J. W. Guo",
    "Wadim Zudilin"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT",
    "math.CA",
    "math.CO",
    "math.QA"
  ],
  "abstract": "By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a \"$q$-microscopic\" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a $q$-analogue of Ramanujan's formula $$ \\sum_{n=0}^\\infty\\frac{\\binom{4n}{2n}{\\binom{2n}{n}}^2}{2^{8n}3^{2n}}\\,(8n+1) =\\frac{2\\sqrt{3}}{\\pi}, $$ of the two supercongruences $$ S(p-1)\\equiv p\\biggl(\\frac{-3}p\\biggr)\\pmod{p^3} \\quad\\text{and}\\quad S\\Bigl(\\frac{p-1}2\\Bigr) \\equiv p\\biggl(\\frac{-3}p\\biggr)\\pmod{p^3}, $$ valid for all primes $p>3$, where $S(N)$ denotes the truncation of the infinite sum at the $N$-th place and $(\\frac{-3}{\\cdot})$ stands for the quadratic character modulo $3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-05T18:50:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "11F33",
      "11Y60",
      "33C20",
      "33D15"
    ],
    "keywords": [
      "supercongruences",
      "microscope",
      "quadratic character modulo",
      "truncated ordinary hypergeometric sums",
      "infinite basic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}