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13
Appell polynomials and their relatives
 Int. Math. Res. Not
, 2004
"... 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 va ..."
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Cited by 17 (1 self)
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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 pseudoorthogonality is given. Another family investigated are the KailathSegall 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 qdeformed case. 1.
qLévy processes
 J. Reine Angew. Math
, 2004
"... ABSTRACT. We continue the investigation of the Lévy processes on a qdeformed 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 ..."
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ABSTRACT. We continue the investigation of the Lévy processes on a qdeformed 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.
Wick’s theorem for q–deformed boson operators
 Journal of Physics A: Mathematical and General
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THESE DE DOCTORAT Université de FrancheComté Ecole doctorale Louis Pasteur Discipline
"... Estimations de normes dans les espaces Lp non commutatifs et applications Directeur de thèse: Christian LE MERDY Soutenue le 25 novembre 2011 devant le jury composé de: ..."
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Estimations de normes dans les espaces Lp non commutatifs et applications Directeur de thèse: Christian LE MERDY Soutenue le 25 novembre 2011 devant le jury composé de:
PRODUCTTYPE NONCOMMUTATIVE POLYNOMIAL STATES
, 811
"... ABSTRACT. In [Ans08a, Ans08b], we investigated monic multivariate noncommutative 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 ..."
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ABSTRACT. In [Ans08a, Ans08b], we investigated monic multivariate noncommutative 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 noncommutative polynomials, namely for various product states. In particular, we introduce a notion of a producttype state on polynomials, which covers all the noncommutative universal products and excludes some other familiar noncommutative products, and which guarantees a number of nice properties for the corresponding polynomials. 1.
Meixner class of noncommutative generalized stochastic processes with freely independent values I. A Characterization
, 2009
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BessisMoussaVillani conjecture and generalized Gaussian random variables
, 2007
"... In this paper we give the solution of BessisMoussaVillani conjecture (BMV) conjecture for the generalized Gaussian random variables G(f) = a(f) + a∗(f), where f is in the real Hilbert space H. The main examples of generalized Gaussian random variables are qGaussian random variables, (−1 ≤ q ≤ 1) ..."
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In this paper we give the solution of BessisMoussaVillani conjecture (BMV) conjecture for the generalized Gaussian random variables G(f) = a(f) + a∗(f), where f is in the real Hilbert space H. The main examples of generalized Gaussian random variables are qGaussian random variables, (−1 ≤ q ≤ 1), related to q −CCR relation and others commutation relations. We will prove that (BMV) conjecture is true for all operators A = G(f), B = G(g); i.e. we will show that the function F (x) = tr(exp(A+ ixB)) is positive definite function on the real line. The case q = 0,i.e. when G(f) are the free Gaussian (Wigner) random variables and the operators A and B are free with respect to the vacuum trace was proved by M.Fannes and D.Petz [23]. 1 Generalized Gaussian Random Variable. Generalized Gaussian random variables, G(f) were introduced in our paper with R.Speicher [16], where the main example was coming from the qCCR relation for q ∈ [−1, 1]: a(f)a∗(g) − qa∗(g)a(f) =< f, g> I,
Appell Polynomials and Their Relatives
"... Let µ be a probability measure on the real line, all of whose moments ..."
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A COMBINATORIAL APPROACH TO MONOTONIC
, 2006
"... 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 ..."
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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 innerproduct bimodule analogues for the monotone and weakly monotone product of Hilbert spaces, an adhoc version of the Central Limit Theorem, an operatorvalued arsine distribution as well as a connection to operatorvalued conditional freeness. 1.
A NONCOMMUTATIVE ANALOGUE OF GAUSSIAN HILBERT SPACES
, 2005
"... 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 AmitsurLevitzki Theorem. ..."
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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 AmitsurLevitzki Theorem. 1.