{
  "id": "1703.01955",
  "version": "v1",
  "published": "2017-03-06T16:28:31.000Z",
  "updated": "2017-03-06T16:28:31.000Z",
  "title": "Arithmetic properties of coefficients of power series expansion of $\\prod_{n=0}^{\\infty}\\left(1-x^{2^{n}}\\right)^{t}$ (with an Appendix by Andrzej Schinzel)",
  "authors": [
    "Maciej Gawron",
    "Piotr Miska",
    "Maciej Ulas"
  ],
  "comment": "34 pages, submitted",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $F(x)=\\prod_{n=0}^{\\infty}(1-x^{2^{n}})$ be the generating function for the Prouhet-Thue-Morse sequence $((-1)^{s_{2}(n)})_{n\\in\\N}$. In this paper we initiate the study of the arithmetic properties of coefficients of the power series expansions of the function $$ F_{t}(x)=F(x)^{t}=\\sum_{n=0}^{\\infty}f_{n}(t)x^{n}. $$ For $t\\in\\N_{+}$ the sequence $(f_{n}(t))_{n\\in\\N}$ is the Cauchy convolution of $t$ copies of the Prouhet-Thue-Morse sequence. For $t\\in\\Z_{<0}$ the $n$-th term of the sequence $(f_{n}(t))_{n\\in\\N}$ counts the number of representations of the number $n$ as a sum of powers of 2 where each summand can have one among $-t$ colors. Among other things, we present a characterization of the solutions of the equations $f_{n}(2^k)=0$, where $k\\in\\N$, and $f_{n}(3)=0$. Next, we present the exact value of the 2-adic valuation of the number $f_{n}(1-2^{m})$ - a result which generalizes the well known expression concerning the 2-adic valuation of the values of the binary partition function introduced by Euler and studied by Churchhouse and others.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-06T16:28:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P81",
      "11P83",
      "11B50"
    ],
    "keywords": [
      "power series expansion",
      "arithmetic properties",
      "andrzej schinzel",
      "coefficients",
      "prouhet-thue-morse sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}