Heterogeneous r-Stirling numbers via boson operators
- 1Kwangwoon Global Education Center, Kwangwoon University, Seoul139-701, Republic of Korea
- 2Department of Mathematics, Sogang University, Seoul121-742, Republic of Korea
- 3Department of Mathematics, Kwangwoon University, Seoul139-701, Republic of Korea
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 ⟨ a+a⟩ n,λ (a+)r and ⟨a+a+r⟩n,λ ⟨ a+a+r⟩ n,λ , 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.
Related
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BibTeX
@article{dmitry2026heterogeneous,
title = {{Heterogeneous r-Stirling numbers via boson operators}},
author = {Dolgy Dmitry V. and Kim Dae San and Kim Taekyun and Lee Hyeonseok and Shahzadi Hira},
journal = {Open Physics},
year = {2026},
month = oct,
doi = {10.1515/phys-2025-0325},
url = {https://doi.org/10.1515/phys-2025-0325},
note = {CC BY},
}
RIS
TY - JOUR
TI - Heterogeneous r-Stirling numbers via boson operators
AU - Dolgy Dmitry V.
AU - Kim Dae San
AU - Kim Taekyun
AU - Lee Hyeonseok
AU - Shahzadi Hira
PY - 2026
DA - 2026/10/26
JO - Open Physics
DO - 10.1515/phys-2025-0325
UR - https://doi.org/10.1515/phys-2025-0325
AB - 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 ⟨ a^+a⟩ _n,λ (a^+)^r and ⟨a+a+r⟩n,λ ⟨ a^+a+r⟩ _n,λ , 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.
SN - 2391-5471
ER -