Meixner Class of Non-commutative Generalized Stochastic Processes with Freely Independent Values II. The Generating Function

Abstract

Let T be an underlying space with a non-atomic measure σ on it. In [Comm. Math. Phys. 292, 99–129 (2009)] the Meixner class of non-commutative generalized stochastic processes with freely independent values, ${\omega=(\omega(t))_{t\in T}}$ , was characterized through the continuity of the corresponding orthogonal polynomials. In this paper, we derive a generating function for these orthogonal polynomials. The first question we have to answer is: What should serve as a generating function for a system of polynomials of infinitely many non-commuting variables? We construct a class of operator-valued functions ${Z=(Z(t))_{t\in T}}$ such that Z(t) commutes with ω(s) for any ${s,t\in T}$. Then a generating function can be understood as ${G(Z,\omega)=\sum_{n=0}^\infty \int_{T^n}P^{(n)}(\omega(t_1),\dots,\omega(t_n))Z(t_1)\dots Z(t_n)}$ ${\sigma(dt_1)\,\dots\,\sigma(dt_n)}$ , where ${P^{(n)}(\omega(t_1),\dots,\omega(t_n))}$ is (the kernel of the) n ^th orthogonal polynomial. We derive an explicit form of G(Z, ω), which has a resolvent form and resembles the generating function in the classical case, albeit it involves integrals of non-commuting operators. We finally discuss a related problem of the action of the annihilation operators ${\partial_t,t \in T}$ . In contrast to the classical case, we prove that the operators ∂ _t related to the free Gaussian and Poisson processes have a property of globality. This result is genuinely infinite-dimensional, since in one dimension one loses the notion of globality.