{
  "abstract": "Let $T(x)=\\prod_{j\\ge 0}(1-x^{2^j})=\\sum_{n\\ge 0}t(n)x^n$ be the Thue--Morse generating function, and write $T(x)^m=\\sum_{n\\ge 0}t_m(n)x^n$ for a positive integer $m$. For a nonzero integer $a$, let $ν_2(a)$ be the exponent of $2$ in $a$ and put $\\operatorname{odd}(a)=a/2^{ν_2(a)}$. For every $s\\ge 1$, we prove that the sequence $(\\operatorname{odd}(t_m(n))\\bmod 2^s)_{n\\ge 0}$ is $2$-automatic when $m=2^r$, $r\\ge 1$, or $m=3\\cdot 2^r$, $r\\ge 2$. In both families, $ν_2(t_m(n))=ν_2\\binom{n+m-1}{m-1}$. The valuation identity for $m=2^r$ is known; our proof recovers it and also establishes the automaticity assertion. For $m=6$ we prove $ν_2(t_6(n))=ν_2\\binom{n+5}{5}+\\mathbf{1}_{n\\equiv 3\\pmod 4}$ and show that its odd parts modulo every $2^s$ are $2$-automatic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16966",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhao Shen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/050yvpqjcv9zb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16966",
  "fulltext_url": null,
  "id": "050yvpqjcv9zb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16966",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7213404659",
  "pdf_url": "https://arxiv.org/pdf/2609.16966",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/050yvpqjcv9zb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16966",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16966"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Powers of the Thue--Morse Series: 2-Adic Valuations and Automatic Odd Parts",
  "title_text": "Powers of the Thue--Morse Series: 2-Adic Valuations and Automatic Odd Parts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/050yvpqjcv9zb",
  "venue": "arXiv",
  "versions": []
}