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A closed form for the generalized Stirling numbers of boson ordering

Abstract

The generalized Stirling numbers with parameters (-1, 2; r), which govern boson operator orderings, have the closed form conjectured by Maier for every integer r, once the first binomial coefficient is read as the generalized binomial coefficient.

Definition 1.1 (The numbers S-hat).

Formalization. D5/S3/Quantum/FockSpace/BosonOrderingStirlingClosedForm.stirlingHat (✓ std3).

Citation. Robert S. Maier (2024). Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers. DOI: 10.48550/arXiv.2308.10332. URL: https://arxiv.org/abs/2308.10332v4.

Commentary.

Theorem 4.1 of the source: for alpha = -1 and beta = 2 the number S-hat(n, k; r) is the k-th forward difference at x = 0 of the rising factorial (2x + r)(2x + r + 1) … (2x + r + n - 1).

Definition 1.2 (The conjectured sum).

Formalization. D5/S3/Quantum/FockSpace/BosonOrderingStirlingClosedForm.conjectureSum (✓ std3).

Citation. Robert S. Maier (2024). Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers. DOI: 10.48550/arXiv.2308.10332. URL: https://arxiv.org/abs/2308.10332v4.

Commentary.

The sum over integers j from floor((2 - r)/2) to floor((n + 2 - r)/2) of the generalized binomial coefficient C(n - j, n - k), times n! and the ordinary binomial coefficient C(n + 1, 2j + r - 1), whose lower index lies between 0 and n + 1 in this range.

Definition 1.3 (The conjecture).

Formalization. D5/S3/Quantum/FockSpace/BosonOrderingStirlingClosedForm.claim (✓ std3).

Citation. Robert S. Maier (2024). Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers. DOI: 10.48550/arXiv.2308.10332. URL: https://arxiv.org/abs/2308.10332v4.

Commentary.

Conjecture 5.3 of the source, for all n, all k <= n and all integers r. The source notes that the upper argument n - j may be negative; with the convention that a binomial coefficient with negative upper argument vanishes, the identity fails already at n = 1, k = 1, r = -1.

Theorem 1.4 (Proof of the conjecture).

Proof. Machine-checked in Lean as D5/S3/Quantum/FockSpace/BosonOrderingStirlingClosedForm.result (✓ std3). ∎

Resolves. Problems/maier-2024-boson-ordering-stirling-closed-form (proved) by D5/S3/Quantum/FockSpace/BosonOrderingStirlingClosedForm.result.

Source. Repository-derived.

Acknowledgement. Robert S. Maier (2024). Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers. DOI: 10.48550/arXiv.2308.10332. URL: https://arxiv.org/abs/2308.10332v4.

Commentary.

Extend the sum to all integers j; the terms outside the range vanish. Both sides f(n, k, r) satisfy f(n, k + 1, r) = f(n, k, r + 2) - f(n, k, r) for k < n: on the left because forward differences commute with the shift x -> x + 1, which sends r to r + 2; on the right by shifting j and Pascal’s rule for the generalized binomial coefficient. At k = 0 the left side is the rising factorial of r, and both sides satisfy f(n + 1, 0, r) - f(n + 1, 0, r - 1) = (n + 1) f(n, 0, r): on the left because the rising factorials of r and r - 1 share n factors, on the right by Pascal’s rule applied twice and a shift of j. At r = 1 both sides equal n!, that is f(n, 0, 1) = n! for every n, so induction on n and then on the integer r settles k = 0, and induction on k settles every k <= n.

References