Generalized Segal-Bargmann transform for Poisson distribution revisited

For $\alpha>0$ and $\sigma > 0$, we consider the following probability distribution on $\alpha\mathbb N_0$: $\pi_{\alpha,\sigma} = \exp \big(- \frac{\sigma}{{\alpha}^2}\big) \sum_{n=0}^{\infty} \frac{1}{n!} \big(\frac{\sigma}{{\alpha}^2}\big)^n {\delta}_{\alpha n}$, where $\delta_y$ denotes the Dirac measure with mass at $y$. For $\alpha=1$, $\pi_{1,\sigma}$ is the Poisson distribution with parameter $\sigma$. Furthermore, the centered probability distribution $\tilde \pi_{\alpha,\sigma} = \exp \big(- \frac{\sigma}{{\alpha}^2}\big) \sum_{n=0}^{\infty} \frac{1}{n!} \big(\frac{\sigma}{{\alpha}^2}\big)^n {\delta}_{\alpha n-\sigma/\alpha}$ weakly converges to $\mu_\sigma$ as $\alpha\to0$. Here $\mu_\sigma$ is the Gaussian distribution with mean zero and variance $\sigma$. Let $(c_n)_{n=0}^\infty$ be the monic polynomial sequence that is orthogonal with respect to the measure $\mu_{\alpha,\sigma}$. In particular, for $\alpha=1$, $(c_n)_{n=0}^\infty$ is a sequence of Charlier polynomials. Let $\mathbb F_\sigma(\mathbb C)$ denote the Bargmann space of all entire functions $f(z)=\sum_{n=0}^\infty f_nz^n$ with $f_n \in \mathbb C$ satisfying $\sum_{n=0}^{\infty} {| f_n |}^2 \, n! \, \sigma^n < \infty$. The generalized Segal--Bargmann transform associated with the measure $\pi_{\alpha,\sigma}$ is a unitary operator $\mathcal S:L^2(\alpha\mathbb N_0,\pi_{\alpha,\sigma})\to \mathbb F_\sigma(\mathbb C)$ that satisfies $(\mathcal Sc_n)(z)=z^n$ for $n\in\mathbb N_0$. We present some new results related to the operator $\mathcal S$. In particular, we observe how the study of $\mathcal S$ naturally leads to the normal ordering in the Weyl algebra.