{
  "abstract": "We give a self-contained derivation of the closed form of the power sums $S_m(n)=\\sum_{i=1}^{n}(a_0+(i-1)d)^m$ in which the Bernoulli numbers enter as constants of integration rather than as prescribed coefficients. The derivation rests on an elementary fact: a sum whose terms are polynomial in the index is the restriction to the integers of a unique polynomial, which satisfies the defining difference equation identically and may therefore be differentiated and integrated exactly. Differentiating lowers the power by one; integrating back introduces a single constant, equal to the slope $S_m'(0)$ of the polynomial at the origin and fixed by the value $S_m(1)=a_0^m$. This yields a closed form for $S_m(n)$ in terms of the constants and a recurrence that determines them, obtained without induction on the exponent and without generating functions. With the Bernoulli numbers defined by their classical recurrence, we identify the constants as $d^mB_m(a_0/d)$; the progression starting at $0$ reproduces that recurrence exactly. In the normalized case they are $B_m(1)$, so $C_1=B_1(1)=+\\tfrac12$, a value that reflects the summation range; a reflection identity shows, without generating functions, that the odd constants beyond the first vanish. The coefficients of the recurrence are scaled binomial coefficients from a single row of Pascal's triangle, and the same approach gives closed forms for sequences with polynomial differences. The integral recursion behind the method is known in the normalized case; each result is related to its classical counterpart, and a verified implementation is provided.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18898",
  "author_total": 1,
  "authors": [
    {
      "name": "Ahmed Abdlmohsin Abdlsahib"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/89nhwzjybasj9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18898",
  "fulltext_url": null,
  "id": "89nhwzjybasj9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18898",
  "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.GM"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7222222230",
  "pdf_url": "https://arxiv.org/pdf/2610.18898",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/89nhwzjybasj9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.18898",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18898"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Advanced Mathematical Identities",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Symbolic Differentiation_Summation Framework for Power Sums of Arithmetic Sequences",
  "title_text": "A Symbolic Differentiation_Summation Framework for Power Sums of Arithmetic Sequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/89nhwzjybasj9",
  "venue": "arXiv",
  "versions": []
}