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16
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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Cited by 9 (5 self)
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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 and Fluctuations of Random Matrices
, 2005
"... Abstract. In this paper we establish a connection between the fluctuations of Wishart random matrices, shifted Chebyshev polynomials, and planar diagrams whose linear span form a basis for the irreducible representations of the annular TemperlyLieb algebra. 1. ..."
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Cited by 8 (2 self)
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Abstract. In this paper we establish a connection between the fluctuations of Wishart random matrices, shifted Chebyshev polynomials, and planar diagrams whose linear span form a basis for the irreducible representations of the annular TemperlyLieb algebra. 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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Cited by 8 (1 self)
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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.
AskeyWilson polynomials, quadratic harnesses and martingales
 sumbitted) arxiv.org/abs/0812.0657, 2008. □ W̷LODEK BRYC AND JACEK WESO̷LOWSKI
"... Abstract. We use orthogonality measures of AskeyWilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. AskeyWilson polynomials are orthogonal martingale polynomials for these processes. ..."
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Cited by 2 (2 self)
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Abstract. We use orthogonality measures of AskeyWilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. AskeyWilson polynomials are orthogonal martingale polynomials for these processes.
On the orthogonal polynomials associated with a Lévy process
 Ann. Probab
"... of all orders. There are two families of orthogonal polynomials associated with X. On one hand, the Kailath–Segall formula gives the relationship between the iterated integrals and the variations of order n of X, and defines a family of polynomials P1(x1), P2(x1,x2),... that are orthogonal with resp ..."
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Cited by 1 (0 self)
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of all orders. There are two families of orthogonal polynomials associated with X. On one hand, the Kailath–Segall formula gives the relationship between the iterated integrals and the variations of order n of X, and defines a family of polynomials P1(x1), P2(x1,x2),... that are orthogonal with respect to the joint law of the variations of X. On the other hand, we can construct a sequence of orthogonal polynomials p σ n(x) with respect to the measure σ 2 δ0(dx) + x 2 ν(dx), where σ 2 is the variance of the Gaussian part of X and ν its Lévy measure. These polynomials are the building blocks of a kind of chaotic representation of the square functionals of the Lévy process proved by Nualart and Schoutens. The main objective of this work is to study the probabilistic properties and the relationship of the two families of polynomials. In particular, the Lévy processes such that the associated polynomials Pn(x1,...,xn) depend on a fixed number of variables are characterized. Also, we give a sequence of Lévy processes that converge in the Skorohod topology to X, such that all variations and iterated integrals of the sequence converge to the variations and iterated integrals of X.
ULTRASPHERICAL TYPE GENERATING FUNCTIONS FOR ORTHOGONAL POLYNOMIALS
, 812
"... Abstract. We characterize the probability distributions of finite all order moments having generating functions for orthogonal polynomials of ultraspherical type. 1. Motivation: Meixner ..."
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Abstract. We characterize the probability distributions of finite all order moments having generating functions for orthogonal polynomials of ultraspherical type. 1. Motivation: Meixner
TOPICS ON MEIXNER FAMILIES
, 812
"... Abstract. We shed the light on the interconnections between different characterizations leading to the classical Meixner family. This allow to set free analogs of both the Sheffer’s and the AlSalam and Chihara’s characterizations in the classical case by the use of the free derivative operator. Th ..."
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Abstract. We shed the light on the interconnections between different characterizations leading to the classical Meixner family. This allow to set free analogs of both the Sheffer’s and the AlSalam and Chihara’s characterizations in the classical case by the use of the free derivative operator. The paper is closed with a discussion of the qdeformed case, q  < 1. 1.
Lowering and raising operators for the free Meixner class of orthogonal polynomials
, 2008
"... We compare some properties of the lowering and raising operators for the classical and free classes of Meixner polynomials on the real line. ..."
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We compare some properties of the lowering and raising operators for the classical and free classes of Meixner polynomials on the real line.
ULTRASPHERICAL TYPE GENERATING FUNCTIONS FOR ORTHOGONAL POLYNOMIALS
, 812
"... Abstract. We characterize, up to a conjecture, probability distributions of finite all order moments with ultraspherical type generating functions for orthogonal polynomials. 1. Motivation: Meixner ..."
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Abstract. We characterize, up to a conjecture, probability distributions of finite all order moments with ultraspherical type generating functions for orthogonal polynomials. 1. Motivation: Meixner