arXiv subjects
Publications and source records attributed to Abdelhay Benmoussa.
We derive a normal ordering formula for the operator \((xI)^n\), where \(I\) denotes the Volterra operator. The resulting coefficients are shown to coincide with the Bessel numbers. We also present two applications, along with a generalization of the main result.
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.
We establish a new identity linking Bernoulli, Stirling (first kind), and Bessel (first kind) numbers: \[ \sum_{k=0}^{n} 2^{\,n-k}\,s(n,k)\,B_k \;=\; \sum_{k=0}^{n} b(n,k)\,\frac{(-1)^k\,k!}{k+1}. \] This parallels the classical Stirling--Bernoulli relation \[ B_n = \sum_{k=0}^{n} S(n,k)\,\frac{(-1)^k\,k!}{k+1}, \] replacing $S(n,k)$ with $s(n,k)$ and $b(n,k)$, and thus revealing a new structural connection among these families of numbers.