Normal Ordering in the Algebra Generated by x and I and a Combinatorial Generalization of Bessel Numbers

2025/11/29 by Abdelhay Benmoussa, Benmoussa, Abdelhay
Mathematics · #11B83 #26A36 #44A45 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic and Geometric Analysis #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2512.00416

openalex publication_date 2025/11/29 · openalex created_date 2025/12/03 · openalex updated_date 2026/07/28

Abstract

We investigate the algebra generated by the operators x and I = ∫₀^x, which satisfy the commutation relation [I,x] = Ix - xI = - I². We develop a combinatorial framework for the normal ordering of words in this algebra and show that any word can be written in the form w = ∑_i,j c(i,j) xⁱ I^j, where the coefficients c(i,j) are signed integers. Focusing on powers of the operator (xI)ⁿ, we demonstrate that the corresponding coefficients coincide with the classical Bessel numbers (OEIS A001498). We further extend this analysis to powers of the generalized operators (x^λI^δ)ⁿ and, finally, provide an explicit normal-ordered expression for an arbitrary word.

Citations

• Reconstruction from k-decks for graphs with maximum degree 2
• Complexity Results in Graph Reconstruction

Related

• Rook numbers and the normal ordering problem
• Combinatorics of Boson Normal Ordering: the Dobinski Formula Revisited
• Normal ordering associated with λ-Stirling numbers inλ-Shift algebra
• Normal ordered grammars
• Combinatorial proofs of inverse relations and log-concavity for Bessel numbers