{
  "annotation_id": "ce7461a3-942a-4418-9494-ae77bbd166ce",
  "date_created": "2026-02-17T05:53:08.667000Z",
  "date_modified": "2026-02-17T05:53:08.667000Z",
  "file_hash": "360f927da4fc2c069baae4ddbf4ce3e8c63c7933629a7d7f2474e5d0821c08ea",
  "private": false,
  "record": {
    "abstract": "In this paper we derive generalizations of different properties of monic polynomial families of binomial type, i.e. families of monic polynomials, for which the binomial theorem holds $$ p_n(\\alpha+\\beta)=\\sum_{k=0}^n \\left(\\vphantom{\\bigg|}\\genfrac{}{}{0pt}{0}{n}{k}\\right) p_k(\\alpha)p_{n-k}(\\beta) $$ Some trivial representations of general \u0027\u0027multiplication\u0027\u0027 and \u0027\u0027derivative\u0027\u0027 operators are derived. In addition we derive a formula for the logarithmic derivative of general monic polynomial $p_n(x)$ which reduces to the formula $$ \\frac{1}{n}\\frac{p_n\u0027(x)}{p_n(x)} =\\left(x+\\frac{1}{\\varphi\u0027(y)}\\left(\\frac{d}{dy}-n\\mathrm{L}\\right)\\right)^{-1}\\cdot\\left.\\frac{\\varphi(y)}{y\\varphi\u0027(y)}~\\right|_{y=0} $$ derived by the author in binomial case, when the generating function of $p_n(x)$ equals to $e^{x\\varphi(y)}$.",
    "arxiv_id": "2601.07029",
    "authors": [
      "Danil Krotkov"
    ],
    "categories": [
      "math.NT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "On families of monic polynomials",
    "url": "https://arxiv.org/abs/2601.07029",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "e7a6c9b4-064b-4bce-ad93-67f47e55c85b",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}