Generalized Segal-Bargmann transform for Poisson distribution revisited

For $\alpha >0$ and $\sigma >0$, we consider the following probability distribution on $\alpha {\mathbb{N}}_0$: ${\pi }_{\alpha ,\sigma }=\exp (-\frac{\sigma }{{\alpha }^2}){\sum }_{n=0}^{\mathrm{\infty }}\frac{1}{n!}(\frac{\sigma }{{\alpha }^2}{)}^n{\delta }_{\alpha n}$, 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 ${\tilde{\pi }}_{\alpha ,\sigma }=\exp (-\frac{\sigma }{{\alpha }^2}){\sum }_{n=0}^{\mathrm{\infty }}\frac{1}{n!}(\frac{\sigma }{{\alpha }^2}{)}^n{\delta }_{\alpha n-\sigma \mathrm{/}\alpha }$ weakly converges to ${\mu }_{\sigma }$ as $\alpha \to 0$. Here ${\mu }_{\sigma }$ is the Gaussian distribution with mean zero and variance $\sigma$. Let $(c_n{)}_{n=0}^{\mathrm{\infty }}$ be the monic polynomial sequence that is orthogonal with respect to the measure ${\mu }_{\alpha ,\sigma }$. In particular, for $\alpha =1$, $(c_n{)}_{n=0}^{\mathrm{\infty }}$ is a sequence of Charlier polynomials. Let ${\mathbb{F}}_{\sigma }(\mathbb{C})$ denote the Bargmann space of all entire functions $f(z)={\sum }_{n=0}^{\mathrm{\infty }}{f}_n{z}^n$ with $f_n\in \mathbb{C}$ satisfying ${\sum }_{n=0}^{\mathrm{\infty }}{\mathrm{\mid }{f}_n\mathrm{\mid }}^{2}n!{\sigma }^n<\mathrm{\infty }$. 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 $(\mathcal{S}{c}_n)(z)=z^n$ 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.