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

We investigate the algebra generated by the operators $x$ and $\mathrm{I}={\int }_0^x$, which satisfy the commutation relation $$[\mathrm{I},x]=\mathrm{I}x-x\mathrm{I}=-{\mathrm{I}}^2\mathrm{.}$$ 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.