{
  "abstract": "This paper investigates the properties and operator-theoretic representations of the heterogeneous r-Stirling numbers of the first and second kind. Building upon the recent introduction of heterogeneous Stirling sequences, we extend these concepts to the r-Stirling case and characterize them using the framework of boson creation (a +) and annihilation (a) operators. We derive explicit normal ordering results for the operators ⟨a+a⟩n,λ(a+)r ${\\langle {a}^{+}a\\rangle }_{n,\\lambda }{({a}^{+})}^{r}$ and ⟨a+a+r⟩n,λ ${\\langle {a}^{+}a+r\\rangle }_{n,\\lambda }$ , showing that their coefficients are given by the heterogeneous r-Stirling numbers of the second kind. Furthermore, we establish several new recurrence relations for both kinds of heterogeneous r-Stirling numbers using the underlying Heisenberg–Weyl algebra. These results provide a unified combinatorial and algebraic representation of these generalized sequences.",
  "abstract_shown": true,
  "affiliations": [
    "Kwangwoon Global Education Center, Kwangwoon University, Seoul139-701, Republic of Korea",
    "Department of Mathematics, Sogang University, Seoul121-742, Republic of Korea",
    "Department of Mathematics, Kwangwoon University, Seoul139-701, Republic of Korea"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 5,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Dolgy Dmitry V."
    },
    {
      "affs": [
        1
      ],
      "name": "Kim Dae San"
    },
    {
      "affs": [
        2
      ],
      "name": "Kim Taekyun"
    },
    {
      "affs": [
        2
      ],
      "name": "Lee Hyeonseok"
    },
    {
      "affs": [
        2
      ],
      "name": "Shahzadi Hira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/avbkaqgdq270v.bib",
  "cited_by": 0,
  "doi": "10.1515/phys-2025-0325",
  "fulltext_url": "https://doi.org/10.1515/phys-2025-0325",
  "id": "avbkaqgdq270v",
  "issns": [
    "2391-5471"
  ],
  "landing_url": "https://doi.org/10.1515/phys-2025-0325",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "lcc",
      "QC1-999"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223067789",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": "De Gruyter",
  "ris": "https://research.draconitic.com/paper/avbkaqgdq270v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T13:47:17Z",
      "native_id": "17a177ee637144578da2d88456d5b909",
      "source": "doaj",
      "url": "https://doi.org/10.1515/phys-2025-0325"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2607|Advanced Combinatorial Mathematics",
      "subfield": "combinatorics"
    }
  ],
  "title": "Heterogeneous r-Stirling numbers via boson operators",
  "title_text": "Heterogeneous r-Stirling numbers via boson operators",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T13:47:17Z",
  "url": "https://research.draconitic.com/paper/avbkaqgdq270v",
  "venue": "Open Physics",
  "versions": []
}