bibkey: maier2024bosonordering authors: Robert S. Maier year: 2024 title: “Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers” doi: 10.48550/arXiv.2308.10332 url: https://arxiv.org/abs/2308.10332v4 claim: “The paper expresses normal-ordering identities for boson creation and annihilation operators through the generalized Stirling numbers of Hsu and Shiue and associated generalized Eulerian numbers, gives closed forms for several parameter pairs, and conjectures (Conjecture 5.3) a closed form for the case alpha = -1, beta = 2 with a generalized binomial coefficient.” strata_touched:

• D5/S3/Quantum/FockSpace/BosonOrderingStirlingClosedForm license: citation-only triage: anchor

Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers

Maier works with the Hsu–Shiue generalized Stirling numbers S_{n,k}(α, β; r), 0 ≤ k ≤ n, and the rescaled numbers Ŝ_{n,k} = β^k k! S_{n,k}, defined in §4 by the expansion

(βx + r)^{n, α} = Σ_{k=0}^{n} Ŝ_{n,k}(α, β; r) C(x, k),

where (y)^{n, α} = y(y − α) ⋯ (y − (n − 1)α) is the generalized falling factorial. Theorem 4.1 gives the equivalent finite sum Ŝ_{n,k} = Σ_{x=0}^{k} (−1)^{k−x} C(k, x) (βx + r)^{n, α}. These numbers connect orderings of words in the boson operators of the Weyl–Heisenberg algebra studied in §3 and §6. For α = −1, β = 2 the factorial is the rising factorial y(y + 1) ⋯ (y + n − 1). After two closed-form theorems in §5, the paper states, as found heuristically:

Conjecture 5.3. For all r ∈ ℤ, Ŝ_{n,k}(−1, 2; r) = Σ_{j=⌊(2−r)/2⌋}^{⌊(n+2−r)/2⌋} C(n − j, n − k) n! C(n + 1, 2j + r − 1),

and remarks that the upper argument n − j of the first binomial coefficient may be negative.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2308.10332
• URL: https://arxiv.org/abs/2308.10332v4
• Version and location: arXiv:2308.10332v4 (2024-02-09; journal version Adv. in Appl. Math. 156 (2024), Paper No. 102678, not read), source file main.tex: §4 for the definition of Ŝ_{n,k}(α, β; r) and Theorem 4.1; §5 for Conjecture 5.3 (the conjecture environment numbered with the theorem counter within sections).