Normal Ordering in the Algebra Generated by $x$ and $\mathrm{I}$ and a Combinatorial Generalization of Bessel Numbers
Abdelhay Benmoussa
TL;DR
The paper studies normal ordering in the algebra generated by $x$ and $\mathrm{I}=\int_0^x$ with $[\mathrm{I},x]=-\mathrm{I}^2$, and develops a combinatorial framework that expresses any word as $w=\sum_{i,j} c(i,j) x^i \mathrm{I}^j$ with integer coefficients. It proves that the normal-ordered form of $(x\mathrm{I})^n$ involves the Bessel numbers, and extends the approach to generalized operators $(x^\lambda \mathrm{I}^\delta)^n$ with a corresponding recurrence. An explicit normal-ordered form for arbitrary words is provided via a structured multi-sum, along with a method to invert nested sums to a monotone cumulative-sum representation. These results connect operator normal ordering with classical combinatorial sequences and yield algorithmic tools for symbolic manipulation in this non-Weyl algebra.
Abstract
We investigate the algebra generated by the operators $x$ and $\mathrm{I} = \int_0^x$, which satisfy the commutation relation \[ [\mathrm{I},x] = \mathrm{I}x - x\mathrm{I} = - \mathrm{I}^2. \] 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 \[ w = \sum_{i,j} c(i,j) \, x^i \mathrm{I}^j, \] where the coefficients $c(i,j)$ are signed integers. Focusing on powers of the operator $(x\mathrm{I})^n$, 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 $(x^λ\mathrm{I}^δ)^n$ and, finally, provide an explicit normal-ordered expression for an arbitrary word.