Fock space associated to Coxeter group of type B

In this article we construct a generalized Gaussian process coming from Coxeter groups of type B. It is given by creation and annihilation operators on an $(\alpha ,q)$-Fock space, which satisfy the commutation relation $$b_{\alpha ,q}(x)b_{\alpha ,q}^{\ast }(y)-q{b}_{\alpha ,q}^{\ast }(y)b_{\alpha ,q}(x)=⟨x,y⟩I+\alpha ⟨\bar{x},y⟩q^{2N},$$ where $x,y$ are elements of a complex Hilbert space with a self-adjoint involution $x\mapsto \bar{x}$ and $N$ is the number operator with respect to the grading on the $(\alpha ,q)$-Fock space. We give an estimate of the norms of creation operators. We show that the distribution of the operators $b_{\alpha ,q}(x)+b_{\alpha ,q}^{\ast }(x)$ with respect to the vacuum expectation becomes a generalized Gaussian distribution, in the sense that all mixed moments can be calculated from the second moments with the help of a combinatorial formula related with set partitions. Our generalized Gaussian distribution associates the orthogonal polynomials called the $q$-Meixner-Pollaczek polynomials, yielding the $q$-Hermite polynomials when $\alpha =0$ and free Meixner polynomials when $q=0$.