ABSTRACT. This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials among polynomials in noncommuting variables. They also fit well into the framework of free probability. For the free Appell polynomials, a number of combinatorial and “diagram ” formulas are proven, such as the formulas for their linearization coefficients. An explicit formula for their generating function is obtained. These polynomials are also martingales for free Lévy processes. For more general free Sheffer families, a necessary condition for pseudo-orthogonality is given. Another family investigated are the Kailath-Segall polynomials. These are multivariate polynomials, which share with the Appell polynomials nice combinatorial properties, but are always orthogonal. Their origins lie in the Fock space representations, or in the theory of multiple stochastic integrals. Diagram formulas are proven for these polynomials as well, even in the q-deformed case. 1.

ABSTRACT. We continue the investigation of the Lévy processes on a q-deformed full Fock space started in [1]. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals with respect to diagonal measures, in the style of Nualart and Schoutens. We define stochastic integration with respect to these processes, and calculate their combinatorial stochastic measures. Finally, we show that they generate infinite von Neumann algebras. 1.

Abstract. In this paper combinatorial aspects of normal ordering arbitrary words in the creation and annihilation operators of the q-deformed boson are discussed. In particular, it is shown how by introducing appropriate q-weights for the associated “Feynman diagrams ” the normally ordered form of a general expression in the creation and annihilation operators can be written as a sum over all q-weighted Feynman diagrams, representing Wick’s theorem in the present context.

Abstract. The paper gives analogues of some starting results in the theory of Gaussian Hilbert Spaces for semicircular distributed random variables. The transition from the commutative to the free frame is done considering matrices of increasing dimension and utilizing the Amitsur-Levitzki Theorem. 1.

Abstract. The notion of monotonic independence is considered in a more general frame, similar to the construction of operator-valued free probability. The paper presents constructions for maps with similar properties to the H and K transforms from the literature, semi inner-product bimodule analogues for the monotone and weakly monotone product of Hilbert spaces, an ad-hoc version of the Central Limit Theorem, an operator-valued arcsine distribution as well as a connection to operator-valued conditional freeness. 1.

Abstract. The notion of monotonic independence, introduced by N. Muraki, is considered in a more general frame, similar to the construction of operatorvalued free probability. The paper presents constructions for maps with similar properties to the H and K transforms from the literature, semi inner-product bimodule analogues for the monotone and weakly monotone product of Hilbert spaces, an ad-hoc version of the Central Limit Theorem, an operator-valued arcsine distribution as well as a connection to operator-valued conditional freeness. 1.

Abstract. The notion of monotonic independence, introduced by N. Muraki, is considered in a more general frame, similar to the construction of operatorvalued free probability. The paper presents constructions for maps with similar properties to the H and K transforms from the literature, semi inner-product bimodule analogues for the monotone and weakly monotone product of Hilbert spaces, an ad-hoc version of the Central Limit Theorem, an operator-valued arsine distribution as well as a connection to operator-valued conditional freeness. 1.

ABSTRACT. In [Ans08a, Ans08b], we investigated monic multivariate non-commutative orthogonal polynomials, their recursions, states of orthogonality, and corresponding continued fraction expansions. In this note, we collect a number of examples, demonstrating what these general results look like for the most important states on non-commutative polynomials, namely for various product states. In particular, we introduce a notion of a product-type state on polynomials, which covers all the non-commutative universal products and excludes some other familiar non-commutative products, and which guarantees a number of nice properties for the corresponding polynomials. 1.

class of non-commutative generalized stochastic processes with freely independent values