Free product of generalized Gaussian processes, random matrices and positive definite functions on permutation groups (Mathematical Aspects of Quantum Fields and Related Topics)

The paper deals with the free product of generalized Gaussian process with function t_{b}(V)=b^{H(V)}, where H(V)=n-h(V), h(V) is the number of singletons in a pair-partition Vin mathcal{P}_{2}(2n). Some new combinatorial formulas are presented. Connections with free additive convolutions probability measure on mathbb{R} are also done. Also new positive definite functions on permutations are presented and also it is proved that the function H is norm (on the group S(infty)=cup S(n). Connection with random matrices and positive definite functions on permutations groups are also done

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 $\sigma$ 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 $G(Z,\omega)=\sum_{n=0}^\infty \int_{T^n}P^{(n)}(\omega(t_1),...,\omega(t_n))Z(t_1)...Z(t_n)\sigma(dt_1)...\sigma(dt_n)$, 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