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.