Generalized Segal–Bargmann transform for Poisson distribution revisited

For Formula: see text and Formula: see text, we consider the following probability distribution on Formula: see text: Formula: see text, where Formula: see text denotes the Dirac measure with mass at Formula: see text. For Formula: see text, Formula: see text is the Poisson distribution with parameter Formula: see text. Furthermore, the centered probability distribution Formula: see text weakly converges to Formula: see text as Formula: see text. Here Formula: see text is the Gaussian distribution with mean zero and variance Formula: see text. Let Formula: see text be the monic polynomial sequence that is orthogonal with respect to the measure Formula: see text. In particular, for Formula: see text, Formula: see text is a sequence of Charlier polynomials. Let Formula: see text denote the Bargmann space of all entire functions Formula: see text with Formula: see text satisfying Formula: see text. The generalized Segal–Bargmann transform associated with the measure Formula: see text is a unitary operator Formula: see text that satisfies Formula: see text for Formula: see text. We present some new results related to the operator Formula: see text. In particular, we observe how the study of Formula: see text naturally leads to the normal ordering in the Weyl algebra.