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 (alpha, q)-Fock space, which satisfy the commutation relation b(alpha,q)(x)b*(alpha,q)(y) - qb*(alpha,q()y)b(alpha,q)(x) = < x,y > I + alpha <(x) over bar, y)q(2N), where x, y are elements of a complex Hilbert space with a self-adjoint involution x bar right arrow (x) over bar 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)(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 alpha = 0 and free Meixner polynomials when q = 0. (C) 2015 Elsevier Inc. All rights reserved.