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 $\sigma$ on it. In [{\it Comm.\ Math.\ Phys.}\ {\bf 292} (2009), 99--129] 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 $\omega (s)$ for any $s,t\in T$. Then a generating function can be understood as $G(Z,\omega )={\sum }_{n=0}^{\infty }{\int }_{T^n}{P}^{(n)}(\omega (t_1),...,\omega (t_n))Z(t_1)...Z(t_n)\sigma (d{t}_1)...\sigma (d{t}_n)$, where $P^{(n)}(\omega (t_1),...,\omega (t_n))$ is (the kernel of the) $n$-th orthogonal polynomial. We derive an explicit form of $G(Z,\omega )$, 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 $\di_t$ 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.

References in corpus (4)

• On a class of free Levy laws related to a regression problem
• Orthogonal polynomials with a resolvent-type generating function
• Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization
• Free Meixner states

Cited by in corpus (4)

• Characterizations of some free random variables by properties of conditional moments of third degree polynomials
• Non-commutative stochastic distributions and applications to linear systems theory
• The projection spectral theorem and Jacobi fields
• Some Fock spaces with depth two action