Generalized Segal-Bargmann transform for Poisson distribution revisited

TL;DR

This paper revisits the generalized Segal-Bargmann transform for Poisson and related distributions, exploring its properties, connections to orthogonal polynomials, and implications for the Weyl algebra's normal ordering.

Contribution

It introduces new results on the operator associated with the generalized Segal-Bargmann transform, linking it to orthogonal polynomials and the Weyl algebra's normal ordering.

Findings

The transform is a unitary operator between specific $L^2$ space and Bargmann space.

Connections between the transform and Charlier polynomials are established.

Insights into the normal ordering in the Weyl algebra are derived from the study of the operator.

Abstract

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.