Generalized Segal-Bargmann transform for Poisson distribution revisited
Chadaphorn Kodsueb, Eugene Lytvynov
Abstract
For $α>0$ and $σ> 0$, we consider the following probability distribution on $α\mathbb N_0$: $π_{α,σ} = \exp \big(- \fracσ{α^2}\big) \sum_{n=0}^{\infty} \frac{1}{n!} \big(\fracσ{α^2}\big)^n δ_{αn}$, where $δ_y$ denotes the Dirac measure with mass at $y$. For $α=1$, $π_{1,σ}$ is the Poisson distribution with parameter $σ$. Furthermore, the centered probability distribution $\tilde π_{α,σ} = \exp \big(- \fracσ{α^2}\big) \sum_{n=0}^{\infty} \frac{1}{n!} \big(\fracσ{α^2}\big)^n δ_{αn-σ/α}$ weakly converges to $μ_σ$ as $α\to0$. Here $μ_σ$ is the Gaussian distribution with mean zero and variance $σ$. Let $(c_n)_{n=0}^\infty$ be the monic polynomial sequence that is orthogonal with respect to the measure $μ_{α,σ}$. In particular, for $α=1$, $(c_n)_{n=0}^\infty$ is a sequence of Charlier polynomials. Let $\mathbb F_σ(\mathbb C)$ 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! \, σ^n < \infty$. The generalized Segal--Bargmann transform associated with the measure $π_{α,σ}$ is a unitary operator $\mathcal S:L^2(α\mathbb N_0,π_{α,σ})\to \mathbb F_σ(\mathbb C)$ 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.