{
  "id": "1907.07572",
  "version": "v1",
  "published": "2019-07-17T15:14:40.000Z",
  "updated": "2019-07-17T15:14:40.000Z",
  "title": "Automatic sequences defined by Theta functions and some infinite products",
  "authors": [
    "Shuo Li"
  ],
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $p(x) \\in C(x)$ be a rational function satisfying the condition $p(0)=1$ and $q$ an integer larger than $1$, in this article we will consider the power expansion of the infinite product $$f(x)=\\prod_{s=0}^{\\infty}f(x^{q^{s}})=\\sum_{i=0}^{\\infty}c_ix^i,$$ and study when the sequence $(c_i)_{i \\in \\mathbf{N}}$ is $q$-automatic. The main result is that for given integers $q \\geq 2$ and $d \\geq 0$, there exist finitely many polynomials of degree $d$ defined over the field of rational numbers $\\mathbf{Q}$, such that $\\prod_{s=0}^{\\infty}f(x^{q^{s}})=\\sum_{i=1}^{\\infty}c_ix^i$ is a $q$-automatic power series.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-17T15:14:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "infinite product",
      "theta functions",
      "automatic sequences",
      "automatic power series",
      "integer larger"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}