DOI: 10.1515/phys-2025-0325 ISSN: 2391-5471

Heterogeneous r -Stirling numbers via boson operators

Dmitry V. Dolgy, Dae San Kim, Taekyun Kim, Hyeonseok Lee, Hira Shahzadi
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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.

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