Radial Bargmann representation for the Fock space of type B

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.