Normal Ordering and Bessel Numbers with Integral Operators

TL;DR

This paper derives a normal ordering formula for a specific operator involving the Volterra operator, revealing coefficients that match Bessel numbers, and explores applications and generalizations of this result.

Contribution

It introduces a novel normal ordering formula for \\((xI)^n\\) with coefficients equal to Bessel numbers, including new applications and a generalization.

Findings

Coefficients match Bessel numbers

Derived a normal ordering formula for \\((xI)^n\\)

Presented applications and generalizations

Abstract

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.