2025/08/26 by Kodsueb, Chadaphorn, Lytvynov, Eugene
#30H20 #46E20 #81R10 #81R30 #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2508.19038
For α>0 and σ> 0, we consider the following probability distribution on α\mathbb N0: πα,σ = exp (- \fracσα2) ∑n=0∞ (1)/(n!) (\fracσα2)n δαn, where δy denotes the Dirac measure with mass at y. For α=1, π1,σ is the Poisson distribution with parameter σ. Furthermore, the centered probability distribution πα,σ = exp (- \fracσα2) ∑n=0∞ (1)/(n!) (\fracσα2)n δαn-σ/α weakly converges to μσ as α→0. Here μσ is the Gaussian distribution with mean zero and variance σ. Let (cn)n=0^∞ be the monic polynomial sequence that is orthogonal with respect to the measure μα,σ. In particular, for α=1, (cn)n=0^∞ is a sequence of Charlier polynomials. Let \mathbb Fσ(\mathbb C) denote the Bargmann space of all entire functions f(z)=∑n=0^∞ fnzn with fn ∈ \mathbb C satisfying ∑n=0∞ | fn |2 n! σn < ∞. The generalized Segal--Bargmann transform associated with the measure πα,σ is a unitary operator \mathcal S:L2(α\mathbb N0,πα,σ)→ \mathbb Fσ(\mathbb C) that satisfies (\mathcal Scn)(z)=zn for n∈\mathbb N0. 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.