Heterogeneous r -Stirling numbers via boson operators
Year: 2026
Type: article
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.
Source: Open Physics
Institutions: Kwangwoon University, Sogang University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,356
OpenAlex ID: W7223067789