Radial Bargmann representation for the Fock space of type B

TL;DR

This paper derives the radial Bargmann representation for the probability measure associated with the $(eta,q)$-Gaussian process in the type B Fock space, extending known results to new distributions and revealing operator relations.

Contribution

It provides the first radial Bargmann representation for the $ u_{eta,q}$ measure and explores the associated $(eta,q)$-operators' commutation relations.

Findings

Representation of $ u_{eta,q}$ measure established

Extension of $q$-Gaussian and free Meixner distributions

New non-trivial operator commutation relations

Abstract

Let $\nu_{\alpha,q}$ be the probability and orthogonality measure for the $q$-Meixner-Pollaczek orthogonal polynomials, which has appeared in \cite{BEH15} as the distribution of the $(\alpha,q)$-Gaussian process (the Gaussian process of type B) over the $(\alpha,q)$-Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of $\nu_{\alpha,q}$. Our main results cover not only the representation of $q$-Gaussian distribution by \cite{LM95}, but also of $q^2$-Gaussian and symmetric free Meixner distributions on $\mathbb R$. In addition, non-trivial commutation relations satisfied by $(\alpha,q)$-operators are presented.