{
  "id": "0709.4031",
  "version": "v2",
  "published": "2007-09-26T17:28:30.000Z",
  "updated": "2011-03-27T21:39:35.000Z",
  "title": "Infinite products with strongly $B$-multiplicative exponents",
  "authors": [
    "Jean-Paul Allouche",
    "Jonathan Sondow"
  ],
  "comment": "Updated [14] and journal reference",
  "journal": "Annales Univ. Sci. Budapest. Sect. Comput. 28 (2008) 35-53. Errata 32 (2010) 253",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $N_{1,B}(n)$ denote the number of ones in the $B$-ary expansion of an integer $n$. Woods introduced the infinite product $P :=\\prod_{n \\geq 0} (\\frac{2n+1}{2n+2})^{(-1)^{N_{1,2}(n)}}$ and Robbins proved that $P = 1/\\sqrt{2}$. Related products were studied by several authors. We show that a trick for proving that $P^2 = 1/2$ (knowing that $P$ converges) can be extended to evaluating new products with (generalized) strongly $B$-multiplicative exponents. A simple example is $$ \\prod_{n \\geq 0} (\\frac{Bn+1}{Bn+2})^{(-1)^{N_{1,B}(n)}} = \\frac{1}{\\sqrt B}. $$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-03-27T21:39:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "11Y60"
    ],
    "keywords": [
      "infinite product",
      "multiplicative exponents",
      "ary expansion",
      "simple example",
      "related products"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.4031A"
    }
  }
}