Generalized Segal-Bargmann transform for Poisson distribution revisited

Abstract

For $\hat{I}\pm >0$ and $\ddot{I}ƒ>0$, we consider the following probability distribution on $\hat{I}\pm {\mathbb{N}}_0$: $\ddot{I}{€}_{\hat{I}\pm ,\ddot{I}ƒ}=\exp (-\frac{\ddot{I}}{ƒ}\hat{I}{\pm }^2){\sum }_{n=0}^{\infty }\frac{1}{n!}(\frac{\ddot{I}}{ƒ}\hat{I}{\pm }^2{)}^n\hat{I}{´}_{\hat{I}\pm n}$, where $\hat{I}{´}_y$ denotes the Dirac measure with mass at $y$. For $\hat{I}\pm =1$, $\ddot{I}{€}_{1,\ddot{I}ƒ}$ is the Poisson distribution with parameter $\ddot{I}ƒ$. Furthermore, the centered probability distribution $\tilde{\ddot{I}}{€}_{\hat{I}\pm ,\ddot{I}ƒ}=\exp (-\frac{\ddot{I}}{ƒ}\hat{I}{\pm }^2){\sum }_{n=0}^{\infty }\frac{1}{n!}(\frac{\ddot{I}}{ƒ}\hat{I}{\pm }^2{)}^n\hat{I}{´}_{\hat{I}\pm n-\ddot{I}ƒ/\hat{I}\pm }$ weakly converges to $\hat{I}{¼}_{\ddot{I}}ƒ$ as $\hat{I}\pm \to 0$. Here $\hat{I}{¼}_{\ddot{I}}ƒ$ is the Gaussian distribution with mean zero and variance $\ddot{I}ƒ$. Let $(c_n{)}_{n=0}^{\infty }$ be the monic polynomial sequence that is orthogonal with respect to the measure $\hat{I}{¼}_{\hat{I}\pm ,\ddot{I}ƒ}$. In particular, for $\hat{I}\pm =1$, $(c_n{)}_{n=0}^{\infty }$ is a sequence of Charlier polynomials. Let ${\mathbb{F}}_{\ddot{I}}ƒ(\mathbb{C})$ denote the Bargmann space of all entire functions $f(z)={\sum }_{n=0}^{\infty }{f}_n{z}^n$ with $f_n\in \mathbb{C}$ satisfying ${\sum }_{n=0}^{\infty }{|f_n|}^{2}n!\ddot{I}{ƒ}^n<\infty$. The generalized Segal--Bargmann transform associated with the measure $\ddot{I}{€}_{\hat{I}\pm ,\ddot{I}ƒ}$ is a unitary operator $\mathcal{S}:L^2(\hat{I}\pm {\mathbb{N}}_0,\ddot{I}{€}_{\hat{I}\pm ,\ddot{I}ƒ})\to {\mathbb{F}}_{\ddot{I}}ƒ(\mathbb{C})$ 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.