{
  "id": "1709.04104",
  "version": "v1",
  "published": "2017-09-13T01:37:01.000Z",
  "updated": "2017-09-13T01:37:01.000Z",
  "title": "Infinite products involving binary digit sums",
  "authors": [
    "Samin Riasat"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $(u_n)_{n\\ge 0}$ be the $\\pm1$ Thue-Morse sequence; i.e., $u_n$ is equal to $1$ if the binary expansion of $n$ has an even number of $1$'s, and is equal to $-1$ otherwise. The following identity, originally due to Woods and Robbins, and several of its generalisations are well-known in the literature \\begin{equation*}\\label{WR}\\prod_{n=0}^\\infty\\left(\\frac{2n+1}{2n+2}\\right)^{u_n}=\\frac{1}{\\sqrt 2}.\\end{equation*} But no other such product involving a rational function in $n$ and the sequence $u_n$ seems to be known in closed form. To understand these products in detail we study the function \\begin{equation*}f(b,c)=\\prod_{n=1}^\\infty\\left(\\frac{n+b}{n+c}\\right)^{u_n}.\\end{equation*} We prove some analytical properties of $f$. We also obtain more identities similar to the Woods-Robbins product, such as \\begin{equation*}\\prod_{n=0}^\\infty\\left(\\frac{4n+1}{4n+3}\\right)^{u_n}=\\frac 12\\end{equation*} \\begin{equation*}\\label{}\\prod_{n=0}^\\infty\\left(\\frac{(2n+2)(n+1)}{(2n+3)(n+2)}\\right)^{u_n}=\\frac{1}{\\sqrt 2}.\\end{equation*}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-13T01:37:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "11B83",
      "11B85",
      "68R15"
    ],
    "keywords": [
      "binary digit sums",
      "infinite products",
      "thue-morse sequence",
      "binary expansion",
      "rational function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}