Normal Ordering in the Algebra Generated by $x$ and $\mathrm{I}$ and a Combinatorial Generalization of Bessel Numbers

TL;DR

This paper develops a combinatorial approach to normal ordering in an algebra generated by operators x and I, revealing connections to Bessel numbers and extending to generalized operators.

Contribution

It introduces a combinatorial framework for normal ordering in the algebra generated by x and I, linking coefficients to classical and generalized Bessel numbers.

Findings

Coefficients in normal ordering match classical Bessel numbers.

Extended analysis to generalized operators (x^λ I^δ)^n.

Explicit normal-ordered form for arbitrary words provided.

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 $(x^\lambda \mathrm{I}^\delta)^n$ and, finally, provide an explicit normal-ordered expression for an arbitrary word.