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47
Free analysis questions. I. Duality transform for the coalgebra of ∂X
 B, Int. Math. Res. Not
"... Preliminary Version A duality transform for the coalgebra of the free difference quotient derivationmultiplication of an operator with respect to a free algebra of scalars is constructed. The dual object is realized in an algebra of matricial analytic functions endowed with yet another generalizatio ..."
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Cited by 27 (1 self)
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Preliminary Version A duality transform for the coalgebra of the free difference quotient derivationmultiplication of an operator with respect to a free algebra of scalars is constructed. The dual object is realized in an algebra of matricial analytic functions endowed with yet another generalization of the difference quotient derivation. 1
On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution
, 2008
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Quadratic harnesses, qcommutations, and orthogonal martingale polynomials
 Trans. Amer. Math. Soc
, 2007
"... Abstract. We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a qcommutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical ..."
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Abstract. We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a qcommutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest. 1.
Conditional moments of qMeixner processes
, 2004
"... Abstract. We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a threeparameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these proce ..."
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Abstract. We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a threeparameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the noncommutative generalizations of the Lévy processes. 1.
Orthogonal polynomials with a resolventtype generating function
, 2004
"... Free Sheffer polynomials are a polynomial family in noncommuting variables with a resolventtype generating function. Among such families, we describe the ones that are orthogonal with respect to a state. Their free cumulant generating functions satisfy a quadratic condition. If this condition is ..."
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Free Sheffer polynomials are a polynomial family in noncommuting variables with a resolventtype generating function. Among such families, we describe the ones that are orthogonal with respect to a state. Their free cumulant generating functions satisfy a quadratic condition. If this condition is linear and the state is tracial, we show that the state is a rotation of a free product state. We also describe interesting examples of nontracial infinitely divisible states with the quadratic property.
N.: Generating functions of Cauchy–Stieltjes type for orthogonal polynomials
 Infin. Dimens. Anal. Quantum Probab. Relat. Top
, 2009
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Free Jacobi processes
, 2006
"... Abstract. In this paper, we define and study free Jacobi processes of parameters λ> 0 and 0 < θ ≤ 1, as the limit of the complex version of the matrix Jacobi process already defined by Y. Doumerc. In the first part, we focus on the stationary case for which we compute the law (that does not de ..."
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Cited by 8 (2 self)
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Abstract. In this paper, we define and study free Jacobi processes of parameters λ> 0 and 0 < θ ≤ 1, as the limit of the complex version of the matrix Jacobi process already defined by Y. Doumerc. In the first part, we focus on the stationary case for which we compute the law (that does not depend on time) and derive, for λ ∈]0, 1] and 1/θ ≥ λ + 1 a free SDE analogous to the classical one. In the second part, we generalize this result under an additional condition. To proceed, we set a recurrence formula for the moments of the process using free stochastic calculus. This will also be used to compute the p. d. e. satisfied by the Cauchy transform of the free Jacobi’s law. 1.
FREE EXPONENTIAL FAMILIES AS KERNEL FAMILIES
, 2008
"... Free exponential families have been previously introduced as a special case of the qexponential family. We show that free exponential families arise also from a procedure analogous to the definition of exponential families by using the CauchyStieltjes kernel (1−θx) −1 instead of the exponential k ..."
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Cited by 8 (0 self)
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Free exponential families have been previously introduced as a special case of the qexponential family. We show that free exponential families arise also from a procedure analogous to the definition of exponential families by using the CauchyStieltjes kernel (1−θx) −1 instead of the exponential kernel e θx. We use this approach to rederive several known results and to study further similarities with exponential families and reproductive exponential models.