Fock space associated to Coxeter groups 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 (α,q)(α,q)-Fock space, which satisfy the commutation relationbα,q(x)bα,q⁎(y)−qbα,q⁎(y)bα,q(x)=〈x,y〉I+α〈x‾,y〉q2N,where x,yx,y are elements of a complex Hilbert space with a self-adjoint involution x↦x¯ and N   is the number operator with respect to the grading on the (α,q)(α,q)-Fock space. We give an estimate of the norms of creation operators. We show that the distribution of the operators bα,q(x)+bα,q⁎(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 is associated to the orthogonal polynomials called the q-Meixner–Pollaczek polynomials, yielding the q  -Hermite polynomials when α=0α=0 and free Meixner polynomials when q=0q=0.