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Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers

Robert S. Maier

TL;DR

The paper addresses boson-string reordering in the Weyl–Heisenberg algebra by developing two complementary expansion schemes whose coefficients are generalized Stirling numbers $S_{n,k}( u)$ and generalized Eulerian numbers $E_{n,k}( u)$. It establishes parametric, closed-form, and recurrences-based methods to compute these coefficients, and shows how they yield new and unified operator-ordering identities that subsume classical normal-ordering results. A key insight is that $S_{n,k}$ and $E_{n,k}$ are binomial transforms of each other and are connected via Vandermonde-type factorizations, with generating functions and special cases linking to Stirling, Lah, and Eulerian number families. The results provide a systematic framework for expressing powers of boson words in terms of alternative words or twistings, with explicit formulas, recurrences, and hypergeometric forms that enhance both quantum-algebraic computations and combinatorial interpretations. Overall, the work extends the toolkit for boson-string orderings and reveals deep connections between operator algebras and generalized combinatorial numbers.

Abstract

Ordering identities in the Weyl-Heisenberg algebra generated by single-mode boson operators are investigated. A boson string composed of creation and annihilation operators can be expanded as a linear combination of other such strings, the simplest example being a normal ordering. The case when each string contains only one annihilation operator is already combinatorially nontrivial. Two kinds of expansion are derived: (i) that of a power of a string $Ω$ in lower powers of another string $Ω'$, and (ii) that of a power of $Ω$ in twisted versions of the same power of $Ω'$. The expansion coefficients are shown to be, respectively, generalized Stirling numbers of Hsu and Shiue, and certain generalized Eulerian numbers. Many examples are given. These combinatorial numbers are binomial transforms of each other, and their theory is developed, emphasizing schemes for computing them: summation formulas, Graham-Knuth-Patashnik (GKP) triangular recurrences, terminating hypergeometric series, and closed-form expressions. The results on the first type of expansion subsume a number of previous results on the normal ordering of boson strings.

Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers

TL;DR

The paper addresses boson-string reordering in the Weyl–Heisenberg algebra by developing two complementary expansion schemes whose coefficients are generalized Stirling numbers and generalized Eulerian numbers . It establishes parametric, closed-form, and recurrences-based methods to compute these coefficients, and shows how they yield new and unified operator-ordering identities that subsume classical normal-ordering results. A key insight is that and are binomial transforms of each other and are connected via Vandermonde-type factorizations, with generating functions and special cases linking to Stirling, Lah, and Eulerian number families. The results provide a systematic framework for expressing powers of boson words in terms of alternative words or twistings, with explicit formulas, recurrences, and hypergeometric forms that enhance both quantum-algebraic computations and combinatorial interpretations. Overall, the work extends the toolkit for boson-string orderings and reveals deep connections between operator algebras and generalized combinatorial numbers.

Abstract

Ordering identities in the Weyl-Heisenberg algebra generated by single-mode boson operators are investigated. A boson string composed of creation and annihilation operators can be expanded as a linear combination of other such strings, the simplest example being a normal ordering. The case when each string contains only one annihilation operator is already combinatorially nontrivial. Two kinds of expansion are derived: (i) that of a power of a string in lower powers of another string , and (ii) that of a power of in twisted versions of the same power of . The expansion coefficients are shown to be, respectively, generalized Stirling numbers of Hsu and Shiue, and certain generalized Eulerian numbers. Many examples are given. These combinatorial numbers are binomial transforms of each other, and their theory is developed, emphasizing schemes for computing them: summation formulas, Graham-Knuth-Patashnik (GKP) triangular recurrences, terminating hypergeometric series, and closed-form expressions. The results on the first type of expansion subsume a number of previous results on the normal ordering of boson strings.
Paper Structure (8 sections, 17 theorems, 80 equations)

This paper contains 8 sections, 17 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. The Weyl--Heisenberg algebra
  4. Generalized Stirling and Eulerian numbers
  5. Stirling-number formulas and combinatorics
  6. Ordering identities (generalized Stirling kind)
  7. Eulerian-number formulas and combinatorics
  8. Ordering identities (generalized Eulerian kind)

Key Result

Lemma 3.1

In $\tilde{W}_{\mathbb{C}}^{\rm{bal}}$, the identities hold for all $n\in\mathbb{N}$ and $\alpha\in\mathbb{C}$. More generally, hold for all $n\in\mathbb{N}$ and $\alpha,r\in\mathbb{C}$, with being a reparametrized statement of the equality between (eq:majorrole2a) and (eq:majorrole2b), which holds for all $n\in\mathbb{N}$ and $L,R\in\mathbb{C}$.

Theorems & Definitions (36)

  • Lemma 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Remark 4.2.1
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • proof
  • ...and 26 more