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

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))_{tin 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))_{tin T}}) such that Z(t) commutes with ω(s) for any ({s,tin 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.